Bias From Pooling Heterogeneous Slopes

Why a Pooled Slope Can Be Misleading

A pooled panel regression reports one slope even when units may respond differently to changes in a regressor. That single number can be useful, but it does not generally represent the response of a typical unit or the simple average of unit-specific responses. The reason is that units contribute different amounts of within-unit information about the regressor.

This tutorial derives the pooled slope for a regression with unit effects, where identification comes from comparing changes within each unit. The key result is that the pooled slope is a weighted average of unit-specific slopes, with weights proportional to each unit’s observed within-unit variation in the regressor. We also distinguish this result from ordinary pooled regression without unit effects, where between-unit differences can affect the coefficient.

For an unbalanced panel, let \(O_i\) be the set of periods in which unit \(i\) is observed, and let \(T_i\) be the number of such periods. Define \(\bar{x}_i\) and \(\bar{y}_i\) as that unit’s observed-period means. Demeaning within unit removes its intercept: \(y_{it}-\bar{y}_i=\beta_i(x_{it}-\bar{x}_i)+(u_{it}-\bar{u}_i)\). The unit’s contribution to the regressor’s within variation is

This quantity is nonnegative and is zero when the observed regressor is constant for that unit. If \(T_i>1\), it can also be written as \((T_i-1)s_{x,i}^2\), where \(s_{x,i}^2\) is the sample variance of the unit’s observed regressor values. Thus the weight reflects both the number of observations and how widely the regressor varies within the unit.

Deriving the Pooled Fixed-Effects Slope

The common fixed-effects slope is the ordinary least-squares slope from the demeaned observations pooled across units. Its numerator adds within-unit cross-products of regressor and outcome; its denominator adds within-unit regressor sums of squares.

Substituting the unit-specific model after demeaning separates the contribution of each slope from the contribution of the errors. This gives the exact finite-sample identity:

The weights are nonnegative and sum to one when the denominator is positive. Since \(\sum_{t\in O_i}(x_{it}-\bar{x}_i)=0\), the error cross-product for unit \(i\) can equivalently use \(u_{it}\) in place of \(u_{it}-\bar{u}_i\). Under the stated strict-exogeneity assumption, the error term has conditional expectation zero given the regressors and unit-specific parameters. Consequently, the conditional expectation of the estimator is the variance-weighted combination \(\sum_i w_i\beta_i\). In a realized sample, the error term need not be exactly zero.

Worked Examples of the Weighting

When Does the Pooled Slope Match an Equal-Unit Average?

For \(N\) units with positive within variation, define the equal-unit average slope as \(\bar{\beta}=N^{-1}\sum_i\beta_i\), and the average within sum of squares as \(\bar{S}_{xx}=N^{-1}\sum_i S_{xx,i}\). Ignoring the sampling error term for the moment, the difference between the variance-weighted slope and the equal-unit average is

To verify the identity, write \(S_{xx,i}=\bar{S}_{xx}+(S_{xx,i}-\bar{S}_{xx})\) and \(\beta_i=\bar{\beta}+(\beta_i-\bar{\beta})\). In the weighted numerator, the terms involving only one deviation sum to zero because \(\sum_i(S_{xx,i}-\bar{S}_{xx})=0\) and \(\sum_i(\beta_i-\bar{\beta})=0\). Subtracting \(\bar{\beta}\sum_i S_{xx,i}\) leaves \(\sum_i(S_{xx,i}-\bar{S}_{xx})(\beta_i-\bar{\beta})\). Dividing by \(\sum_i S_{xx,i}=N\bar{S}_{xx}\) gives the displayed identity.

The numerator on the right is the across-unit covariance, using divisor \(N\), between within variation and slope. If units with larger within variation tend to have larger slopes, the weighted slope tends to exceed the equal-unit average. If their slopes tend to be smaller, it tends to fall below it. The two averages coincide if all slopes are equal, if all units have equal within variation, or more generally if the across-unit covariance is zero.

Why Unit Effects Matter for This Interpretation

The variance-weighted result applies to the regression with unit effects. A regression that pools all observations with only a common intercept does not generally have this interpretation. It can use both within-unit movements and differences between unit means. If unit means of \(x\) are related to unit means of \(y\), that between-unit relationship can change the pooled coefficient even when every unit-specific within slope is known.

Conditions, Scope, and a Common Pitfall

The weighting identity is a statement about the fixed-effects least-squares calculation. Interpreting its error term as having conditional expectation zero requires an exogeneity assumption; if within-unit regressor changes are related to the error, pooling does not remove that source of bias. The panel must also contain positive total within variation, \(\sum_i S_{xx,i}>0\), or the common within slope is not identified.

In an unbalanced panel, observed periods determine both the means used for demeaning and the resulting \(S_{xx,i}\). Therefore the weights describe the units and periods actually represented in the regression. They need not correspond to an equal-unit target, and they can differ from weights based on a different panel composition. Interpreting the estimate for another population or observation pattern requires an explicit argument about that target; the algebra alone does not supply one.

• Pitfall: “The pooled slope is the average effect across units.” This ignores the weights. The fixed-effects pooled slope is generally weighted by \(S_{xx,i}\), not equally across units.
• Correction: State which average is being discussed. For the unit-fixed-effects regression, report the within-variation-weighted interpretation unless additional conditions make it equal to the target of interest.
• Pitfall: “Fixed effects make any pooled slope a within-unit average.” A pooled regression without unit indicators can incorporate between-unit differences.
• Correction: Check whether the fitted regression includes unit effects, and distinguish within-unit variation from between-unit variation before interpreting its coefficient.
• Pitfall: “A unit with more observed periods must have more weight.” Period count matters through the sum of squares, but the within spread of the regressor matters too.
• Correction: Inspect \(S_{xx,i}\), not just \(T_i\). A unit with many observations but little within variation may contribute less slope information than a shorter panel with substantial variation.