Pooled OLS Under Unobserved Unit Heterogeneity

Why Pooled OLS Can Miss the Panel Structure

In a panel, the same units are observed repeatedly. A unit may have a persistent characteristic—such as baseline productivity, management quality, or underlying health—that affects the outcome but is not recorded. If that characteristic is related to the regressors, pooled ordinary least squares (OLS) can attribute some of its effect to the regressor slopes. The resulting problem is omitted-variable bias, even when the regressor and the idiosyncratic error are otherwise unrelated.

The key distinction is between heterogeneity and bias. Unit effects may vary substantially across units without biasing pooled slopes. Bias arises when the part of those effects associated with the regressors has a nonzero linear projection onto the regressors after accounting for the pooled intercept.

Model: For units $i=1,\ldots,N$ observed in periods $t=1,\ldots,T$, consider the static panel equation $$ y_{it}=x_{it}'\beta+\alpha_i+u_{it}. $$ Here $y_{it}$ is the outcome, $x_{it}$ is a $k$-dimensional regressor vector, $\beta$ is the vector of slope parameters, $\alpha_i$ is a time-invariant unit effect, and $u_{it}$ is the idiosyncratic error. Pooled OLS regresses $y_{it}$ on $x_{it}$ with one common intercept, omitting the unit-specific $\alpha_i$.

For the derivations below, suppose the panel is balanced, $T$ is fixed as $N$ grows, and units are sampled independently with moments that obey a law of large numbers. Assume the regressors have finite second moments, and their pooled, intercept-centered second-moment matrix is nonsingular. Also assume the idiosyncratic error satisfies the pooled moment restriction $E_*[(x_{it}-\mu_x)u_{it}]=0$. The notation $E_*$ means an average over units and periods, and $\mu_x=E_*[x_{it}]$. A sufficient, stronger condition is that $u_{it}$ has conditional mean zero given the unit's regressors over all periods and its unit effect. The central issue in this tutorial is whether the analogous moment with $\alpha_i$ is zero.

The Pooled OLS Probability Limit

Let $\hat{\beta}_{P}$ denote the pooled OLS slope estimate, with a common intercept included. Define the pooled centered regressor vector $x_{it}-\mu_x$, and let $\mu_\alpha=E_*[\alpha_i]$. The relevant population moments are

Q=E_*[(x_{it}-\mu_x)(x_{it}-\mu_x)'], \qquad c=E_*[(x_{it}-\mu_x)(\alpha_i-\mu_\alpha)].

The matrix $Q$ describes the variation in the regressors left after fitting the common intercept. The vector $c$ describes how that centered regressor variation co-moves with the unit effect. With $Q$ nonsingular, the pooled OLS probability limit is

\operatorname{plim}\hat{\beta}_{P} =\beta+Q^{-1}c.

This is the omitted-unit-effect bias formula. It concerns the pooled linear projection and does not require $\alpha_i$ to be statistically independent of every regressor. What matters for the pooled slopes is whether its centered linear projection on the included regressors is zero. The formula also makes clear that the consequence of a given association depends on the joint variation among regressors: with multiple regressors, $Q^{-1}$ can distribute the distortion across several slope estimates.

Deriving the Bias Formula

The result follows directly from the pooled least-squares normal equations after removing the common intercept. Let $\bar{x}$ and $\bar{y}$ denote the means over the stacked $NT$ observations. Substituting the model into the centered pooled slope estimator gives

\hat{\beta}_{P} = \left[\sum_{i=1}^{N}\sum_{t=1}^{T}(x_{it}-\bar{x})(x_{it}-\bar{x})'\right]^{-1} \sum_{i=1}^{N}\sum_{t=1}^{T}(x_{it}-\bar{x})(y_{it}-\bar{y}).

Because centering removes constants, substitute $y_{it}=x_{it}'\beta+\alpha_i+u_{it}$ into the numerator. The part involving $x_{it}'\beta$ is the centered regressor cross-product matrix multiplied by $\beta$. The remaining terms are the cross-products with $\alpha_i$ and $u_{it}$. Dividing the cross-products by $NT$, applying the law of large numbers, and using the assumed zero pooled moment for $u_{it}$ yields

\operatorname{plim}\hat{\beta}_{P} = Q^{-1}\{Q\beta+c+0\} = \beta+Q^{-1}c.

The unit effect's pooled mean is absorbed by the common intercept; the centered deviations $\alpha_i-\mu_\alpha$ are not automatically absorbed. If $c=0$, the pooled slopes are consistent under the stated conditions, even if $\alpha_i$ has positive variance. If $c\ne0$, pooled OLS is generally inconsistent for $\beta$ as $N$ grows with $T$ fixed. These are consistency statements under the specified sampling and moment conditions, not claims that every finite-sample estimate must be on one particular side of the true value.

Scalar-regressor result: With one regressor $x_{it}$, the formula reduces to $$ \operatorname{plim}\hat{\beta}_{P} = \beta+ \frac{E_*[(x_{it}-\mu_x)(\alpha_i-\mu_\alpha)]} {E_*[(x_{it}-\mu_x)^2]}. $$ The extra term is the pooled covariance of the regressor with the unit effect divided by the pooled regressor variance.

Worked Examples

Worked Example: A Positive Pooled Slope Bias

Consider hypothetical data from two units observed in two periods. Suppose the outcome obeys $y_{it}=2x_{it}+\alpha_i$, so the true slope is 2 and $u_{it}=0$.

Unit	Period	$x_{it}$	$\alpha_i$	$y_{it}$
A	1	1	1	3
A	2	2	1	5
B	1	3	5	11
B	2	4	5	13

The pooled means are $\bar{x}=2.5$ and $\bar{y}=8$. The centered pooled slope is

\hat{\beta}_{P} = \frac{\sum (x_{it}-\bar{x})(y_{it}-\bar{y})} {\sum (x_{it}-\bar{x})^2} = \frac{(-1.5)(-5)+(-0.5)(-3)+(0.5)(3)+(1.5)(5)} {(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2} = \frac{18}{5} =3.6.

The estimate is 1.6 units of outcome per unit of $x$ above the true slope. The bias formula gives the same result: the centered cross-product of $x$ and $\alpha_i$ is $8$, and the centered sum of squares of $x$ is $5$, so their ratio is $8/5=1.6$. Here the unit with the larger effect also has larger regressor values. The pooled slope incorporates both the structural association and part of the difference between units.

Worked Example: Heterogeneous Effects Without Pooled Slope Bias

Now suppose two hypothetical units have different effects, $\alpha_A=1$ and $\alpha_B=5$, but each follows the same regressor pattern $x=1,3$. Let $y_{it}=2x_{it}+\alpha_i$ and $u_{it}=0$.

Unit	Period	$x_{it}$	$\alpha_i$	$y_{it}$
A	1	1	1	3
A	2	3	1	7
B	1	1	5	7
B	2	3	5	11

The pooled means are $\bar{x}=2$ and $\bar{y}=7$. The centered cross-product for the slope is $\sum(x_{it}-\bar{x})(y_{it}-\bar{y})=4+0+0+4=8$, while $\sum(x_{it}-\bar{x})^2=1+1+1+1=4$. Thus $\hat{\beta}_{P}=8/4=2$, the true slope. The centered cross-product of $x$ and $\alpha_i$ is $2-2-2+2=0$, so the unit effects do not contribute to the pooled slope in this data. There is genuine unit-level heterogeneity, but no pooled linear association between that heterogeneity and the centered regressor.

Worked Example: Which Slopes Are Distorted With Two Regressors?

Consider a hypothetical balanced panel of four units, A–D, observed in two periods. Let $x_1=-1$ for A and B, $x_1=1$ for C and D; let $x_2=-1$ in period 1 and $x_2=1$ in period 2 for every unit. Set $\alpha_A=\alpha_B=0$, $\alpha_C=\alpha_D=2$, and let $y_{it}=2x_{1,it}+3x_{2,it}+\alpha_i$. The eight outcome values, in A–D order for each period, are $-5,-5,1,1$ in period 1 and $1,1,7,7$ in period 2.

Both regressor means are zero, each regressor has pooled second moment 1, and their pooled cross-moment is zero. Therefore $Q$ is the two-by-two identity matrix. The mean unit effect is 1. The centered cross-moment with $x_1$ is 1: $x_1$ and $\alpha_i-1$ both equal $-1$ for A and B and $1$ for C and D. The centered cross-moment with $x_2$ is 0 because the two periods have opposite $x_2$ values for each unit. Hence $c=(1,0)'$, giving

\operatorname{plim}\hat{\beta}_{P} = \begin{pmatrix}2\\3\end{pmatrix} + \begin{pmatrix}1&0\\0&1\end{pmatrix}^{-1} \begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}3\\3\end{pmatrix}.

The pooled coefficient on $x_1$ is shifted from 2 to 3, while the coefficient on $x_2$ remains 3. The difference in unit effects is associated with $x_1$, not $x_2$. This example shows why the multiple-regressor condition is a vector condition: it is the joint centered covariance vector $c$, together with $Q$, that determines the slope distortion.

When Pooled Estimation Remains Consistent

Under the maintained error moment and rank assumptions, the exact population condition for pooled slope consistency is $c=0$. A familiar sufficient condition is that the unit effect is uncorrelated with each centered regressor in the pooled population. Independence of $\alpha_i$ and the entire regressor history is stronger than necessary; zero pooled linear projection is enough for this OLS result. In a multi-regressor model, the direct condition is still $c=0$, rather than an informal judgment that the unit effect “seems unrelated” to a single regressor.

The regressor matrix must also have enough pooled variation to identify the slopes: $Q$ must be nonsingular. Even when $c=0$, pooled OLS does not consistently estimate $\beta$ if the idiosyncratic error is systematically correlated with the included regressors, or if the model's conditional mean is otherwise misspecified. The unit-effect condition is one part of the consistency argument, not a substitute for the other regression assumptions.

Common Mistakes and Why the Distinction Matters

“The unit effect is constant over time, so the common intercept removes it.” A common intercept absorbs the average unit effect, not every unit's deviation from that average. Those deviations can affect the slopes when they covary with regressors.
“Any nonzero variation in $\alpha_i$ biases pooled OLS.” Variation alone does not imply bias. The relevant condition is the centered pooled cross-moment $c$; it can be zero even when unit effects differ.
“Robust or clustered standard errors will fix the pooled coefficient.” Changing the estimated standard error does not change the pooled OLS coefficient or remove its omitted-variable component. Inference adjustments and consistency of the point estimate address different issues.
“A zero association for one regressor guarantees all pooled slopes are consistent.” With multiple regressors, check the full vector $c$ and require $Q$ to be nonsingular.

This distinction matters whenever units differ in persistent, unobserved ways. If the unit effects are orthogonal to the pooled regressors, pooled OLS can still estimate the slopes consistently under the other assumptions. If they are correlated, the pooled coefficient combines the regressor's effect with between-unit differences in the omitted characteristic. Recognizing that source of distortion is the first step in deciding what an appropriate panel estimator must address.

Check Your Understanding

Use the bias formula and the assumptions stated in the tutorial to answer these questions.

In a scalar-regressor model, what pooled moment determines whether the unit effect contributes bias to the pooled slope?
Can pooled OLS slopes be consistent when the variance of $\alpha_i$ is positive? State the relevant condition.
For multiple regressors, what roles do $Q$ and $c$ play in the probability limit?
Why does including a common intercept not generally remove unit-specific deviations from $\alpha_i$?
If $c=0$ but $E_*[(x_{it}-\mu_x)u_{it}]\ne0$, is the stated consistency result justified? Explain.

Unit	Period	\(x_{it}\)	\(\alpha_i\)	\(y_{it}\)
A	1	1	1	3
A	2	2	1	5
B	1	3	5	11
B	2	4	5	13

Unit	Period	\(x_{it}\)	\(\alpha_i\)	\(y_{it}\)
A	1	1	1	3
A	2	3	1	7
B	1	1	5	7
B	2	3	5	11