Common-Factor Error Models in Panels

Why Represent Panel Dependence With Factors?

Panel errors can be correlated across units because many units respond to a shared, possibly unobserved influence. A general dependence diagnostic can reveal that errors move together, but a factor representation makes the structure explicit: units may respond differently to the same latent influence, and their responses are summarized by unit-specific loadings. This distinction matters because the strength of a shared component determines whether its contribution to uncertainty remains important as the number of units grows.

The model below is a representation of the errors, not by itself an estimation method or a claim that the latent factors are causal. Its immediate use is to show how common influences and idiosyncratic errors combine to produce cross-sectional and time dependence.

The Common-Factor Error Model

Definition: A panel error $u_{it}$, for unit $i=1,\ldots,N$ and time $t=1,\ldots,T$, has an $r$-factor representation if $$ u_{it}=\boldsymbol{\lambda}_i'\mathbf{f}_t+e_{it}. $$ Here $\mathbf{f}_t$ is an $r\times 1$ vector of latent common factors at time $t$, $\boldsymbol{\lambda}_i$ is the $r\times 1$ vector of unit $i$'s factor loadings, and $e_{it}$ is the idiosyncratic component. The number $r$ is fixed as $N$ grows in the asymptotic descriptions below.

A factor is common because the same $\mathbf{f}_t$ enters the errors of multiple units. A loading describes how strongly, and in what direction, a unit responds to each factor. For example, with one factor, units with positive and negative loadings respond in opposite directions. The idiosyncratic component captures the part not represented by the common factors; “idiosyncratic” does not automatically mean independent across units or time.

For the covariance calculations, suppose the factors and idiosyncratic errors have zero means, $\mathbf{f}_t$ is uncorrelated with $e_{js}$ for all units and dates, and second moments exist. Write $\boldsymbol{\Gamma}_{ts}=\operatorname{Cov}(\mathbf{f}_t,\mathbf{f}_s)$ and $\omega_{it,js}=\operatorname{Cov}(e_{it},e_{js})$. These conditions give

\operatorname{Cov}(u_{it},u_{js}) =\boldsymbol{\lambda}_i'\boldsymbol{\Gamma}_{ts}\boldsymbol{\lambda}_j +\omega_{it,js}.

The formula applies both to different units at the same date and to observations at different dates. If idiosyncratic errors are uncorrelated across distinct units, then $\omega_{it,jt}=0$ for $i\ne j$, and their same-date covariance comes entirely from the common factors. If factors are serially correlated, they can also transmit dependence across time.

Worked Example: Covariance Across Units and Dates

Suppose a hypothetical one-factor model is $u_{it}=\lambda_i f_t+e_{it}$. Unit A has loading $0.5$, and unit B has loading $-1.5$. Assume $\operatorname{Cov}(f_t,f_s)=0.6$ for two distinct dates $t$ and $s$, and $\operatorname{Cov}(e_{At},e_{Bs})=0.1$. Suppose the factors are uncorrelated with the idiosyncratic errors.

The covariance formula gives $$ \operatorname{Cov}(u_{At},u_{Bs}) =(0.5)(0.6)(-1.5)+0.1=-0.45+0.1=-0.35. $$

Thus the errors have covariance $-0.35$ in the squared units of the errors. The negative factor contribution arises because the two units have loadings with opposite signs; the positive idiosyncratic covariance partly offsets it. A factor model does not require every pair of units to have positive covariance.

Strong and Weak Factors

The number of nonzero factors alone does not say how important their dependence is across a large panel. To describe that importance, consider the $N\times N$ covariance matrix of the common component at a fixed date. Let $\boldsymbol{\Lambda}$ be the $N\times r$ matrix whose $i$th row is $\boldsymbol{\lambda}_i'$, and let $\boldsymbol{\Sigma}_f=\operatorname{Var}(\mathbf{f}_t)$. The common-component covariance matrix is

\boldsymbol{\Lambda}\boldsymbol{\Sigma}_f\boldsymbol{\Lambda}'.

Assume for this comparison that the eigenvalues of $\boldsymbol{\Sigma}_f$ are bounded above and bounded away from zero, and that idiosyncratic cross-sectional covariance is not itself dominant: the largest eigenvalue of the idiosyncratic covariance matrix stays bounded as $N$ grows. These restrictions let us attribute growing eigenvalues to common factors rather than to a separate source of pervasive idiosyncratic dependence.

Definition: A common factor is strong (or pervasive) if its contribution to the common-component covariance matrix has an eigenvalue of order $N$. A factor is weak if the relevant eigenvalue grows more slowly than $N$, for example at order $N^\kappa$ for some $0<\kappa<1$. A factor affecting only a fixed number of units can have an eigenvalue of order one and is a still more localized case. These descriptions concern the factor's aggregate cross-sectional strength, not whether its loading is nonzero for every unit.

For a strong factor, loadings are typically of order one for a nonvanishing share of units, with enough variation in direction that the factor is not eliminated by cancellation. The loading matrix then accumulates information across units, giving a common-component eigenvalue proportional to $N$. A weak factor can have a much smaller effect on each unit, yet still matter in aggregate if it influences many units.

Worked Example: A Strong Factor and a Cross-Sectional Average

Suppose 100 hypothetical units satisfy $u_{it}=f_t+e_{it}$, with $\operatorname{Var}(f_t)=2$. At the date considered, assume the $e_{it}$ are mutually uncorrelated, each with variance $3$, and uncorrelated with $f_t$.

For two distinct units, the error covariance is $$ \operatorname{Cov}(u_{it},u_{jt})=(1)(2)(1)=2. $$ Each unit's error variance is $2+3=5$, so the pairwise correlation is $2/5=0.400$. In matrix form, the common component has covariance $2\mathbf{1}\mathbf{1}'$, where $\mathbf{1}$ is the 100-vector of ones. Its largest eigenvalue is $2(100)=200$, which grows in proportion to $N$; this is a strong factor. Adding the idiosyncratic covariance $3\mathbf{I}$ gives a largest eigenvalue of $203$.

For the cross-sectional average, the common component is still $f_t$, while the average idiosyncratic component has variance $3/100$. Therefore $$ \operatorname{Var}\left(\frac{1}{100}\sum_{i=1}^{100}u_{it}\right) =2+\frac{3}{100}=2.03. $$ The idiosyncratic contribution shrinks as the number of units grows, but the strong common component does not.

What Weak Factors Can Still Do

“Weak” does not mean irrelevant. A factor with a sublinear eigenvalue can generate pairwise covariances that approach zero while its contribution to an aggregate remains larger than the idiosyncratic contribution. The scale of the resulting effect depends on the direction of the loading vector, not just the size of its eigenvalue.

Worked Example: A Weak Factor With Many Small Loadings

For $N=100$ hypothetical units, consider one factor with variance $4$, equal loadings $\lambda_i=N^{-1/4}$, and independent idiosyncratic errors with variance $1$. For these numbers, each loading is approximately $0.3162$.

The common covariance between two distinct errors is $$ \lambda_i\lambda_j\operatorname{Var}(f_t) =N^{-1/2}(4)=\frac{4}{10}=0.400. $$ The common-component covariance matrix is $4N^{-1/2}\mathbf{1}\mathbf{1}'$, so its largest eigenvalue is $4N^{1/2}=40$, not order $N$. It is a weak factor under the eigenvalue definition.

Nevertheless, its contribution to the variance of the cross-sectional average is $40/100=0.400$. The idiosyncratic contribution is $1/100=0.0100$, so the total variance is $0.410$. The weak factor dominates the average's variance in this example even though the factor's eigenvalue grows only as $N^{1/2}$. At larger $N$, each pairwise common covariance decreases, but the factor can still be important relative to an idiosyncratic contribution that shrinks at rate $1/N$.

Why the Covariance Implications Follow

The eigenvalue and average-variance statements follow directly from the loading structure. For one factor with variance $\sigma_f^2$, write $\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_N)'$. The common covariance matrix is $\sigma_f^2\boldsymbol{\lambda}\boldsymbol{\lambda}'$. This rank-one matrix has one nonzero eigenvalue, $\sigma_f^2\boldsymbol{\lambda}'\boldsymbol{\lambda}$, because multiplying it by $\boldsymbol{\lambda}$ gives that eigenvalue times $\boldsymbol{\lambda}$. The variance contribution to the cross-sectional average follows by applying the averaging vector $\mathbf{1}/N$ on both sides:

\operatorname{Var}\left(\frac{1}{N}\sum_{i=1}^{N}\lambda_i f_t\right) =\frac{\sigma_f^2}{N^2}\left(\sum_{i=1}^{N}\lambda_i\right)^2.

If $\lambda_i=1$ for every unit, the eigenvalue is $\sigma_f^2N$ and the average's common variance is $\sigma_f^2$. If instead $\lambda_i=N^{(\kappa-1)/2}$ for every unit, where $0<\kappa<1$, then

\sigma_f^2\boldsymbol{\lambda}'\boldsymbol{\lambda} =\sigma_f^2N^\kappa, \qquad \frac{\sigma_f^2}{N^2}\left(\sum_{i=1}^{N}\lambda_i\right)^2 =\sigma_f^2N^{\kappa-1}.

The second eigenvalue is sublinear in $N$, while the common contribution to the average's variance decreases at rate $N^{\kappa-1}$. For $0<\kappa<1$, that rate is slower than the $1/N$ rate of independent, equal-variance idiosyncratic errors. Thus a weak factor can dominate the variance of an average without being strong in the eigenvalue sense. With unequal or offsetting loadings, the average may instead be less exposed; the eigenvalue measures dependence in the factor's direction, not necessarily in the direction of the simple average.

Interpretation, Scope, and Common Pitfalls

A common-factor representation is useful because it separates two questions: what covariance is generated by shared latent movements, and what dependence remains in the idiosyncratic errors? It also clarifies why having many units does not necessarily eliminate uncertainty in a cross-sectional average. A strong shared component can persist in the average, while a weak component may shrink more slowly than independent noise.

The representation does not identify a unique set of factors and loadings without additional normalization: for any invertible matrix $\mathbf{H}$, the same common component can be written with transformed factors $\mathbf{H}^{-1}\mathbf{f}_t$ and loadings $\mathbf{H}'\boldsymbol{\lambda}_i$. Nor does statistical co-movement establish that a factor is a particular economic shock. The model describes a covariance structure; interpreting factors substantively requires further evidence.

Mistake: “Pairwise covariances are small, so cross-sectional dependence is negligible.” Correction: Many small covariances can accumulate. Inspect the aggregate covariance structure and its eigenvalue growth, as well as the implications for the statistic of interest.
Mistake: “A weak factor cannot affect inference.” Correction: Weakness means sublinear eigenvalue growth, not zero impact. The factor can dominate the variance of an average relative to idiosyncratic noise.
Mistake: “Idiosyncratic means independent.” Correction: It means the component is not represented by the specified common factors. Additional cross-unit or serial dependence can remain unless ruled out by explicit assumptions.
Mistake: “A fitted factor must be a causal common shock.” Correction: A factor representation is a statistical decomposition. Causal interpretation is not implied by the covariance equations.

The factor model therefore gives a language for describing the scale and geometry of panel dependence. Whether the factors are observed or latent, and how to estimate a regression that is robust to them, are separate questions from the covariance representation developed here.

Check Your Understanding

Use the definitions and covariance results above to answer the following.

In $u_{it}=\boldsymbol{\lambda}_i'\mathbf{f}_t+e_{it}$, what role does the loading vector play?
Suppose a one-factor model has $\operatorname{Var}(f_t)=3$, $\lambda_i=2$, and $\lambda_j=-1$. If the two idiosyncratic errors are uncorrelated, what is $\operatorname{Cov}(u_{it},u_{jt})$?
What eigenvalue order distinguishes a strong factor from a weak factor in the definitions used here?
Why can a factor with eigenvalue order $N^{1/2}$ matter more to a cross-sectional average than independent idiosyncratic errors?
Does a common-factor representation alone establish that a latent factor is a causal economic shock? Explain.