单向效应

1 Fixed Effect Model

Consider the one-way error component regression model

\[ Y_{i t}=X_{i t}^{\prime} \beta+u_{i}+\varepsilon_{i t} \]

Define the mean of a variable for a given individual as

\[ \bar{Y}_{i}=\frac{1}{T_{i}} \sum_{t \in S_{i}} Y_{i t} . \]

We call this the individual-specific mean since it is the mean of a given individual. Contrarywise, some authors call this the time-average or time-mean since it is the average over the time periods.

Subtracting the individual-specific mean from the variable we obtain the deviations

\[ \dot{Y}_{i t}=Y_{i t}-\bar{Y}_{i} . \]

This is known as the within transformation.

Some algebra may also be useful. We can write the individual-specific mean as \(\bar{Y}_{i}=\left(\mathbf{1}_{i}^{\prime} \mathbf{1}_{i}\right)^{-1} \mathbf{1}_{i}^{\prime} \boldsymbol{Y}_{i}\). Stacking the observations for individual \(i\) we can write the within transformation using the notation

\[ \begin{aligned} \dot{\boldsymbol{Y}}_{i} &=\boldsymbol{Y}_{i}-\mathbf{1}_{i} \bar{Y}_{i} \\ &=\boldsymbol{Y}_{i}-\mathbf{1}_{i}\left(\mathbf{1}_{i}^{\prime} \mathbf{1}_{i}\right)^{-1} \mathbf{1}_{i}^{\prime} \boldsymbol{Y}_{i} \\ &=\boldsymbol{M}_{i} \boldsymbol{Y}_{i} \end{aligned} \]

where \(\boldsymbol{M}_{i}=\boldsymbol{I}_{i}-\mathbf{1}_{i}\left(\mathbf{1}_{i}^{\prime} \mathbf{1}_{i}\right)^{-1} \mathbf{1}_{i}^{\prime}\) is the individual-specific demeaning operator. Notice that \(\boldsymbol{M}_{i}\) is an idempotent matrix.

Similarly for the regressors we define the individual-specific means and demeaned values:

\[ \begin{aligned} \bar{X}_{i} &=\frac{1}{T_{i}} \sum_{t \in S_{i}} X_{i t} \\ \dot{X}_{i t} &=X_{i t}-\bar{X}_{i} \\ \dot{\boldsymbol{X}}_{i} &=\boldsymbol{M}_{i} \boldsymbol{X}_{i} . \end{aligned} \]

We can alternatively write this in vector notation. Applying the demeaning operator \(\boldsymbol{M}_{i}\) to (17.16) we

\[ \dot{\boldsymbol{Y}}_{i}=\dot{\boldsymbol{X}}_{i} \beta+\dot{\boldsymbol{\varepsilon}}_{i} . \]

2 Fixed Effects Estimator

Consider least squares applied to the demeaned equation (17.21) or equivalently (17.22). This is

\[ \begin{aligned} \widehat{\beta}_{\mathrm{fe}} &=\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \dot{X}_{i t} \dot{X}_{i t}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \dot{X}_{i t} \dot{Y}_{i t}\right) \\ &=\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{Y}}_{i}\right) \\ &=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \boldsymbol{M}_{i} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \boldsymbol{M}_{i} \boldsymbol{Y}_{i}\right) \end{aligned} \]

This is known as the fixed-effects or within estimator of \(\beta\).

Let \(\Sigma_{i}=\mathbb{E}\left[\boldsymbol{\varepsilon}_{i} \boldsymbol{\varepsilon}_{i}^{\prime} \mid \boldsymbol{X}_{i}\right]\) denote the \(T_{i} \times T_{i}\) conditional covariance matrix of the idiosyncratic errors. The variance of \(\widehat{\beta}_{\mathrm{fe}}\) is

\[ \boldsymbol{V}_{\mathrm{fe}}=\operatorname{var}\left[\widehat{\beta}_{\mathrm{fe}} \mid \boldsymbol{X}\right]=\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \Sigma_{i} \dot{\boldsymbol{X}}_{i}\right)\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)^{-1} \]

3 Fixed Effects Covariance Matrix Estimation

First consider estimation of the classical covariance matrix \(\boldsymbol{V}_{\mathrm{fe}}^{0}\) as defined in (17.27). This is

\[ \widehat{\boldsymbol{V}}_{\mathrm{fe}}^{0}=\widehat{\sigma}_{\varepsilon}^{2}\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1} \]

with

\[ \widehat{\sigma}_{\varepsilon}^{2}=\frac{1}{n-N-k} \sum_{i=1}^{n} \sum_{t \in S_{i}} \widehat{\varepsilon}_{i t}^{2}=\frac{1}{n-N-k} \sum_{i=1}^{n} \widehat{\boldsymbol{\varepsilon}}_{i} \widehat{\boldsymbol{\varepsilon}}_{i} . \]

The \(N+k\) degree of freedom adjustment is motivated by the dummy variable representation. You can verify that \(\widehat{\sigma}_{\varepsilon}^{2}\) is unbiased for \(\sigma_{\varepsilon}^{2}\) under assumptions (17.18), (17.25) and (17.26). See Exercise 17.8.

Notice that the assumptions (17.18), (17.25), and (17.26) are identical to (17.5)-(17.7) of Assumption 17.1. The assumptions (17.8)-(17.10) are not needed. Thus the fixed effect model weakens the random effects model by eliminating the assumptions on \(u_{i}\) but retaining those on \(\varepsilon_{i t}\).

The classical covariance matrix estimator (17.36) for the fixed effects estimator is valid when the errors \(\varepsilon_{i t}\) are homoskedastic and serially uncorrelated but is invalid otherwise. A covariance matrix estimator which allows \(\varepsilon_{i t}\) to be heteroskedastic and serially correlated across \(t\) is the cluster-robust covariance matrix estimator, clustered by individual

\[ \widehat{\boldsymbol{V}}_{\mathrm{fe}}^{\text {cluster }}=\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \widehat{\boldsymbol{\varepsilon}}_{i} \widehat{\boldsymbol{\varepsilon}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1} \]

where \(\widehat{\boldsymbol{\varepsilon}}_{i}\) as the fixed effects residuals as defined in (17.23). (17.38) was first proposed by Arellano (1987). As in (4.55) \(\widehat{V}_{\text {fe }}^{\text {cluster }}\) can be multiplied by a degree-of-freedom adjustment. The adjustment recommended by the theory of C. Hansen (2007) is

\[ \widehat{\boldsymbol{V}}_{\mathrm{fe}}^{\text {cluster }}=\left(\frac{N}{N-1}\right)\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \widehat{\boldsymbol{\varepsilon}}_{i} \widehat{\boldsymbol{\varepsilon}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1} \]

and that corresponding to \((4.55)\) is

\[ \widehat{\boldsymbol{V}}_{\mathrm{fe}}^{\text {cluster }}=\left(\frac{n-1}{n-N-k}\right)\left(\frac{N}{N-1}\right)\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1}\left(\sum_{i=1}^{N} \dot{\boldsymbol{X}}_{i}^{\prime} \widehat{\boldsymbol{\varepsilon}}_{i} \widehat{\boldsymbol{\varepsilon}}_{i}^{\prime} \dot{\boldsymbol{X}}_{i}\right)\left(\dot{\boldsymbol{X}}^{\prime} \dot{\boldsymbol{X}}\right)^{-1} \text {. } \]

These estimators are convenient because they are simple to apply and allow for unbalanced panels.

4 Between Estimator

The between estimator is calculated from the individual-mean equation (17.20)

\[ \bar{Y}_{i}=\bar{X}_{i}^{\prime} \beta+u_{i}+\bar{\varepsilon}_{i} . \]

Estimation can be done at the level of individuals or at the level of observations. Least squares applied to (17.41) at the level of the \(N\) individuals is

\[ \widehat{\beta}_{\mathrm{be}}=\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{Y}_{i}\right) . \]

Least squares applied to (17.41) at the level of observations is

\[ \widetilde{\beta}_{\mathrm{be}}=\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \sum_{t \in S_{i}} \bar{X}_{i} \bar{Y}_{i}\right)=\left(\sum_{i=1}^{N} T_{i} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} T_{i} \bar{X}_{i} \bar{Y}_{i}\right) . \]

In balanced panels \(\widetilde{\beta}_{\mathrm{be}}=\widehat{\beta}_{\text {be }}\) but they differ on unbalanced panels. \(\widetilde{\beta}_{\mathrm{be}}\) equals weighted least squares applied at the level of individuals with weight \(T_{i}\).

Under the random effects assumptions (Assumption 17.1) \(\widehat{\beta}_{\text {be }}\) is unbiased for \(\beta\) and has variance

\[ \boldsymbol{V}_{\mathrm{be}}=\operatorname{var}\left[\widehat{\beta}_{\mathrm{be}} \mid \boldsymbol{X}\right]=\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1}\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{X}_{i}^{\prime} \sigma_{i}^{2}\right)\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1} \]

where

\[ \sigma_{i}^{2}=\operatorname{var}\left[u_{i}+\bar{\varepsilon}_{i}\right]=\sigma_{u}^{2}+\frac{\sigma_{\varepsilon}^{2}}{T_{i}} \]

is the variance of the error in (17.41). When the panel is balanced the variance formula simplifies to

\[ \boldsymbol{V}_{\mathrm{be}}=\operatorname{var}\left[\widehat{\beta}_{\mathrm{be}} \mid \boldsymbol{X}\right]=\left(\sum_{i=1}^{N} \bar{X}_{i} \bar{X}_{i}^{\prime}\right)^{-1}\left(\sigma_{u}^{2}+\frac{\sigma_{\varepsilon}^{2}}{T}\right) . \]

Instead, its primary application is to construct an estimate of \(\sigma_{u}^{2}\). First, consider estimation of

\[ \sigma_{b}^{2}=\frac{1}{N} \sum_{i=1}^{N} \sigma_{i}^{2}=\sigma_{u}^{2}+\frac{1}{N} \sum_{i=1}^{N} \frac{\sigma_{\varepsilon}^{2}}{T_{i}}=\sigma_{u}^{2}+\frac{\sigma_{\varepsilon}^{2}}{\bar{T}} \]

where \(\bar{T}=N / \sum_{i=1}^{N} T_{i}^{-1}\) is the harmonic mean of \(T_{i}\). (In the case of a balanced panel \(\bar{T}=T\).) A natural estimator of \(\sigma_{b}^{2}\) is

\[ \widehat{\sigma}_{b}^{2}=\frac{1}{N-k} \sum_{i=1}^{N} \widehat{e}_{b i}^{2} . \]

where \(\widehat{e}_{b i}=\bar{Y}_{i}-\bar{X}_{i}^{\prime} \widehat{\beta}_{\text {be }}\) are the between residuals. (Either \(\widehat{\beta}_{\text {be }}\) or \(\widetilde{\beta}_{\text {be }}\) can be used.)

From the relation \(\sigma_{b}^{2}=\sigma_{u}^{2}+\sigma_{\varepsilon}^{2} / \bar{T}\) and (17.42) we can deduce an estimator for \(\sigma_{u}^{2}\). We have already described an estimator \(\widehat{\sigma}_{\varepsilon}^{2}\) for \(\sigma_{\varepsilon}^{2}\) in (17.37) for the fixed effects model. Since the fixed effects model holds under weaker conditions than the random effects model, \(\widehat{\sigma}_{\varepsilon}^{2}\) is valid for the latter as well. This suggests the following estimator for \(\sigma_{u}^{2}\)

\[ \widehat{\sigma}_{u}^{2}=\widehat{\sigma}_{b}^{2}-\frac{\widehat{\sigma}_{\varepsilon}^{2}}{\bar{T}} . \]

To summarize, the fixed effect estimator is used for \(\widehat{\sigma}_{\varepsilon}^{2}\), the between estimator for \(\widehat{\sigma}_{b}^{2}\), and \(\widehat{\sigma}_{u}^{2}\) is constructed from the two.

It is possible for (17.43) to be negative. It is typical to use the constrained estimator

\[ \widehat{\sigma}_{u}^{2}=\max \left[0, \widehat{\sigma}_{b}^{2}-\frac{\widehat{\sigma}_{\varepsilon}^{2}}{\bar{T}}\right] . \]

(17.44) is the most common estimator for \(\sigma_{u}^{2}\) in the random effects model.

5 Random Effects

It implies that the vector of errors \(\boldsymbol{e}_{i}\) for individual \(i\) has the covariance structure

\[ \begin{aligned} \mathbb{E}\left[\boldsymbol{e}_{i} \mid \boldsymbol{X}_{i}\right] &=0 \\ \mathbb{E}\left[\boldsymbol{e}_{i} \boldsymbol{e}_{i}^{\prime} \mid \boldsymbol{X}_{i}\right] &=\mathbf{1}_{i} \mathbf{1}_{i}^{\prime} \sigma_{u}^{2}+\boldsymbol{I}_{i} \sigma_{\varepsilon}^{2} \\ &=\left(\begin{array}{cccc} \sigma_{u}^{2}+\sigma_{\varepsilon}^{2} & \sigma_{u}^{2} & \cdots & \sigma_{u}^{2} \\ \sigma_{u}^{2} & \sigma_{u}^{2}+\sigma_{\varepsilon}^{2} & \cdots & \sigma_{u}^{2} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{u}^{2} & \sigma_{u}^{2} & \cdots & \sigma_{u}^{2}+\sigma_{\varepsilon}^{2} \end{array}\right) \\ &=\sigma_{\varepsilon}^{2} \Omega_{i}, \end{aligned} \]

The random effects regression model is

\[ Y_{i t}=X_{i t}^{\prime} \beta+u_{i}+\varepsilon_{i t} \]

or \(\boldsymbol{Y}_{i}=\boldsymbol{X}_{i} \beta+\mathbf{1}_{i} u_{i}+\boldsymbol{\varepsilon}_{i}\) where the errors satisfy Assumption 17.1.

Given the error structure the natural estimator for \(\beta\) is GLS. Suppose \(\sigma_{u}^{2}\) and \(\sigma_{\varepsilon}^{2}\) are known. The GLS estimator of \(\beta\) is

\[ \widehat{\beta}_{\mathrm{gls}}=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{Y}_{i}\right) . \]

A feasible GLS estimator replaces the unknown \(\sigma_{u}^{2}\) and \(\sigma_{\varepsilon}^{2}\) with estimators. See Section \(17.15\).

Thus \(\widehat{\beta}_{\text {gls }}\) is conditionally unbiased for \(\beta\). The conditional variance of \(\widehat{\beta}_{\text {gls }}\) is

\[ \boldsymbol{V}_{\mathrm{gls}}=\left(\sum_{i=1}^{n} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1} \sigma_{\varepsilon}^{2} \]

Under the assumption that the random effects model is a useful approximation but not literally true then we may consider a cluster-robust covariance matrix estimator such as

\[ \widehat{\boldsymbol{V}}_{\mathrm{gls}}=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \widehat{\boldsymbol{e}}_{i} \widehat{\boldsymbol{e}}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)\left(\sum_{i=1}^{n} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1} \]

where \(\widehat{\boldsymbol{e}}_{i}=\boldsymbol{Y}_{i}-\boldsymbol{X}_{i} \widehat{\beta}_{\mathrm{gls}}\). This may be re-scaled by a degree of freedom adjustment if desired.

6 Feasible GLS of random effects

The random effects estimator can be written as

\[ \widehat{\beta}_{\mathrm{re}}=\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{X}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \boldsymbol{X}_{i}^{\prime} \Omega_{i}^{-1} \boldsymbol{Y}_{i}\right)=\left(\sum_{i=1}^{N} \widetilde{\boldsymbol{X}}_{i}^{\prime} \widetilde{\boldsymbol{X}}_{i}\right)^{-1}\left(\sum_{i=1}^{N} \widetilde{\boldsymbol{X}}_{i}^{\prime} \widetilde{\boldsymbol{Y}}_{i}\right) \]

where \(\widetilde{\boldsymbol{X}}_{i}=\Omega_{i}^{-1 / 2} \boldsymbol{X}_{i}\) and \(\widetilde{\boldsymbol{Y}}_{i}=\Omega_{i}^{-1 / 2} \boldsymbol{Y}_{i}\). It is instructive to study these transformations.

Define \(\boldsymbol{P}_{i}=\mathbf{1}_{i}\left(\mathbf{1}_{i}^{\prime} \mathbf{1}_{i}\right)^{-1} \mathbf{1}_{i}^{\prime}\) so that \(\boldsymbol{M}_{i}=\boldsymbol{I}_{i}-\boldsymbol{P}_{i}\). Thus while \(\boldsymbol{M}_{i}\) is the within operator, \(\boldsymbol{P}_{i}\) can be called the individual-mean operator since \(\boldsymbol{P}_{i} \boldsymbol{Y}_{i}=\mathbf{1}_{i} \bar{Y}_{i}\). We can write

\[ \Omega_{i}=\boldsymbol{I}_{i}+\mathbf{1}_{i} \mathbf{1}_{i}^{\prime} \sigma_{u}^{2} / \sigma_{\varepsilon}^{2}=\boldsymbol{I}_{i}+\frac{T_{i} \sigma_{u}^{2}}{\sigma_{\varepsilon}^{2}} \boldsymbol{P}_{i}=\boldsymbol{M}_{i}+\rho_{i}^{-2} \boldsymbol{P}_{i} \]

where

\[ \rho_{i}=\frac{\sigma_{\varepsilon}}{\sqrt{\sigma_{\varepsilon}^{2}+T_{i} \sigma_{u}^{2}}} . \]

Since the matrices \(\boldsymbol{M}_{i}\) and \(\boldsymbol{P}_{i}\) are idempotent and orthogonal we find that \(\Omega_{i}^{-1}=\boldsymbol{M}_{i}+\rho_{i}^{2} \boldsymbol{P}_{i}\) and

\[ \Omega_{i}^{-1 / 2}=\boldsymbol{M}_{i}+\rho_{i} \boldsymbol{P}_{i}=\boldsymbol{I}_{i}-\left(1-\rho_{i}\right) \boldsymbol{P}_{i} . \]

Therefore the transformation used by the GLS estimator is

\[ \tilde{\boldsymbol{Y}}_{i}=\left(\boldsymbol{I}_{i}-\left(1-\rho_{i}\right) \boldsymbol{P}_{i}\right) \boldsymbol{Y}_{i}=\boldsymbol{Y}_{i}-\left(1-\rho_{i}\right) \mathbf{1}_{i} \bar{Y}_{i} \]

which is a partial within transformation.

The transformation as written depends on \(\rho_{i}\) which is unknown. It can be replaced by the estimator

\[ \widehat{\rho}_{i}=\frac{\widehat{\sigma}_{\varepsilon}}{\sqrt{\widehat{\sigma}_{\varepsilon}^{2}+T_{i} \widehat{\sigma}_{u}^{2}}} \]

where the estimators \(\widehat{\sigma}_{\varepsilon}^{2}\) and \(\widehat{\sigma}_{u}^{2}\) are given in (17.37) and (17.44). We obtain the feasible transformations

\[ \widetilde{\boldsymbol{Y}}_{i}=\boldsymbol{Y}_{i}-\left(1-\widehat{\rho}_{i}\right) \mathbf{1}_{i} \bar{Y}_{i} \]

and

\[ \widetilde{\boldsymbol{X}}_{i}=\boldsymbol{X}_{i}-\left(1-\widehat{\rho}_{i}\right) \mathbf{1}_{i} \bar{X}_{i}^{\prime} . \]

The feasible random effects estimator is (17.45) using (17.49) and (17.50).

In the previous section we noted that it is possible for \(\widehat{\sigma}_{u}^{2}=0\). In this case \(\widehat{\rho}_{i}=1\) and \(\widehat{\beta}_{\text {re }}=\widehat{\beta}_{\text {pool }}\).

What this shows is the following. The random effects estimator (17.45) is least squares applied to the transformed variables \(\widetilde{\boldsymbol{X}}_{i}\) and \(\widetilde{\boldsymbol{Y}}_{i}\) defined in (17.50) and (17.49). When \(\widehat{\rho}_{i}=0\) these are the within transformations, so \(\widetilde{\boldsymbol{X}}_{i}=\dot{\boldsymbol{X}}_{i}, \widetilde{\boldsymbol{Y}}_{i}=\dot{\boldsymbol{Y}}_{i}\), and \(\widehat{\beta}_{\mathrm{re}}=\widehat{\beta}_{\mathrm{fe}}\) is the fixed effects estimator. When \(\widehat{\rho}_{i}=1\) the data are untransformed \(\widetilde{\boldsymbol{X}}_{i}=\boldsymbol{X}_{i}, \widetilde{\boldsymbol{Y}}_{i}=\boldsymbol{Y}_{i}\), and \(\widehat{\beta}_{\mathrm{re}}=\widehat{\beta}_{\text {pool }}\) is the pooled estimator. In general, \(\widetilde{\boldsymbol{X}}_{i}\) and \(\widetilde{\boldsymbol{Y}}_{i}\) can be viewed as partial within transformations.

Recalling the definition \(\widehat{\rho}_{i}=\widehat{\sigma}_{\varepsilon} / \sqrt{\widehat{\sigma}_{\varepsilon}^{2}+T_{i} \widehat{\sigma}_{u}^{2}}\) we see that when the idiosyncratic error variance \(\widehat{\sigma}_{\varepsilon}^{2}\) is large relative to \(T_{i} \widehat{\sigma}_{u}^{2}\) then \(\widehat{\rho}_{i} \approx 1\) and \(\widehat{\beta}_{\text {re }} \approx \widehat{\beta}_{\text {pool. }}\). Thus when the variance estimates suggest that the individual effect is relatively small the random effect estimator simplifies to the pooled estimator. On the other hand when the individual effect error variance \(\widehat{\sigma}_{u}^{2}\) is large relative to \(\widehat{\sigma}_{\varepsilon}^{2}\) then \(\widehat{\rho}_{i} \approx 0\) and \(\widehat{\beta}_{\mathrm{re}} \approx \widehat{\beta}_{\mathrm{fe}}\). Thus when the variance estimates suggest that the individual effect is relatively large the random effect estimator is close to the fixed effects estimator.