The PANEL Procedure

Balanced Panels

Variance Covariance of Dummy Variables with No Intercept
Variance Covariance of Dummy Variables with Intercept

Assume that the data are balanced (for example, all cross sections have T observations). Then you can write the following:

$\tilde{\mi {y}}_\mi {it} = \mi {y_\mi {it}} - \bar{\mi {y}}_\mi {i \cdot } - \bar{\mi {y}}_\mi {\cdot t} + \bar{\bar{\mi {y}}}$

$\tilde{\mi {\mb {x}}}_\mi {it} = \mi {\mb {x} _\mi {it}} - \bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {\cdot t} + \bar{\bar{\mi {\mb {x}}}}$

where the symbols:

$\mi {y_\mi {it}}$ and $\mi {\mb {x} _\mi {it}}$ are the dependent variable (a scalar) and the explanatory variables (a vector whose columns are the explanatory variables not including a constant), respectively

$\bar{\mi {y}}_\mi {i \cdot }$ and $\bar{\mi {\mb {x}}}_\mi {i \cdot }$ are cross section means

$\bar{\mi {y}}_\mi {\cdot t}$ and $\bar{\mi {\mb {x}}}_\mi {\cdot t}$ are time means

$\bar{\bar{\mi {y}}}$ and $\bar{\bar{\mi {\mb {x}}}}$ are the overall means

The two-way fixed-effects model is simply a regression of $\tilde{\mi {y}}_\mi {it}$ on $\tilde{\mi {\mb {x}}}_\mi {it}$ . Therefore, the two-way ${\beta }$ is given by:

$\tilde{{\beta }}_{s}= \left(\tilde{\mi {\mb {X}}}^{}\tilde{\mi {\mb {X}}} \right)^{-1} \tilde{\mi {\mb {X}}}^{}\tilde{\mb {y}}$

The calculations of cross section dummy variables, time dummy variables, and intercepts follow in a fashion similar to that used in the one-way model.

First, you obtain the net cross-sectional and time effects. Denote the cross-sectional effects by $\gamma$ and the time effects by $\alpha$ . These effects are calculated from the following relations:

$\hat{\gamma }_{i} = \left(\bar{\mi {y}}_\mi {i \cdot }- \bar{\bar{\mi {y}}} \right) - \tilde{{\beta }}_{s}\left( \bar{\mi {x}}_\mi {i \cdot }- \bar{\bar{\mi {x}}} \right)$

$\hat{\alpha }_{t} = \left(\bar{\mi {y}}_{\cdot \mi {t}}- \bar{\bar{\mi {y}}} \right) - \tilde{{\beta }}_{s}\left( \bar{\mi {x}}_{\cdot \mi {t}}- \bar{\bar{\mi {x}}} \right)$

Denote the cross-sectional dummy variables and time dummy variables with the superscript C and T. Under the NOINT option the following equations give the dummy variables:

$D_ i^{C} = \hat{\gamma }_{i} + \hat{\alpha }_{T}$

$D_ t^{T} = \hat{\alpha }_{t}- \hat{\alpha }_{T}$

When an intercept is specified, the equations for dummy variables and intercept are:

$D_ i^{C} = \hat{\gamma }_{i}- \hat{\gamma }_{N}$

$D_ t^{T} = \hat{\alpha }_{t}- \hat{\alpha }_{T}$

$\mr {Intercept }= \hat{\gamma }_{N} + \hat{\alpha }_{T}$

The sum of squared errors is:

$\mr {SSE}= \sum _\mi {i = 1} ^\mi {N} \sum _\mi {t = 1} ^\mi {T_\mi {i}} (y_\mi {it} - \gamma _\mi {i}-\alpha _\mi {t} - \mb {X} _{s}\tilde{{\beta }}_{s})^{2}$

The estimated error variance is:

$\hat{{\sigma }}_{{\epsilon }}^{2}= \mr {SSE }/ (\mi {M}-\mi {N}-\mi {T}-(\mi {K} -1))$

With or without a constant, the variance covariance matrix of ${\tilde{\beta }}_{s}$ is given by:

$\mr {Var}\left[{\tilde{\beta }}_{s}\right] = \hat{{\sigma }}_{{\epsilon }}^{2}(\tilde{\mb {X} }^{}_{s} \tilde{\mb {X} }_{s})^{-1}$

Variance Covariance of Dummy Variables with No Intercept

The variances and covariances of the dummy variables are given with the NOINT specification as follows:

$\displaystyle \mr {Var}\left(D_\emph {i} ^{C}\right)$	$\displaystyle =$	$\displaystyle \hat{\sigma }_{\epsilon }^{2} \left(\frac{1}{T} + \frac{1}{N} - \frac{1}{NT} \right)$
$\displaystyle$	$\displaystyle +$	$\displaystyle \left(\bar{\mi {\mb {x}}}_{\mi {i} \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {i \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)$

$\mr {Var}\left(D_\mi {t} ^{T}\right) = \frac{2\hat{\sigma }_{\epsilon }^{2}}{\mi {N}} +\left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)$

$\displaystyle \mr {Cov}\left(D_\emph {i} ^{C},D_\emph {j} ^{C}\right)$	$\displaystyle =$	$\displaystyle \hat{\sigma }_{\epsilon }^{2} \left(\frac{1}{N} - \frac{1}{NT} \right)$
$\displaystyle$	$\displaystyle +$	$\displaystyle \left(\bar{\mi {\mb {x}}}_\mi {i \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {j \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)$

$\mr {Cov}\left(D_\mi {t} ^{T},D_\mi {u} ^{T}\right) = \frac{\hat{\sigma }_{\epsilon }^{2}}{N} + \left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - {\bar{\mi {\mb {x}}}}_{\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi {\mb {x}}}_\mi {\cdot u} - \bar{\mi {\mb {x}}}_{\cdot T}\right)$

$\mr {Cov}\left(D_\mi {i} ^{C},D_\mi {t} ^{T}\right) = -\frac{\hat{\sigma }_{\epsilon }^{2}}{N} +\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_{\cdot T}\right)$

$\mr {Cov}\left( D_\mi {i} ^ C,\beta \right) = -\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right]$

$\mr {Cov}\left( D_\mi {i} ^ T,\beta \right) = -\left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right]$

Variance Covariance of Dummy Variables with Intercept

The variances and covariances of the dummy variables are given when the intercept is included as follows:

$\displaystyle \mr {Var}\left(D_\mi {i} ^{C}\right)$	$\displaystyle = \frac{2\hat{\sigma }_{\epsilon }^{2}}{T} +\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)$
$\displaystyle \mr {Var}\left(D_\mi {t} ^{T}\right)$	$\displaystyle = \frac{2\hat{\sigma }_{\epsilon }^{2}}{\mi {N}} +\left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)$
$\displaystyle \mr {Var}\left(\mr {Intercept}\right)$	$\displaystyle = \hat{\sigma }_{\epsilon }^{2} \left(\frac{1}{T} + \frac{1}{N} - \frac{1}{NT} \right) +\left(\bar{\mi {\mb {x}}}_\mi {N \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot T} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {N \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot T} - \bar{\bar{\mi {\mb {x}}}}\right)$
$\displaystyle \mr {Cov}\left(D_\mi {i} ^{C},D_\mi {j} ^{C}\right)$	$\displaystyle = \frac{\hat{\sigma }_{\epsilon }^{2}}{T} +\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {j \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)$
$\displaystyle \mr {Cov}\left(D_\mi {t} ^{T},D_\mi {u} ^{T}\right)$	$\displaystyle = \left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi {\mb {x}}}_\mi {\cdot u} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)$
$\displaystyle \mr {Cov}\left(D_\mi {i} ^{C},\mr {Intercept}\right)$	$\displaystyle =-\left(\frac{\hat{\sigma }_{\epsilon }^{2}}{T}\right) +\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)^{} \mr {Var}\left({\tilde{\beta }}_{s}\right) \left(\bar{\mi {\mb {x}}}_\mi {N \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot T} - \bar{\bar{\mi {\mb {x}}}}\right)$
$\displaystyle \mr {Cov}\left(D_\mi {t} ^{T},\mr {Intercept}\right)$	$\displaystyle =-\left(\frac{\hat{\sigma }_{\epsilon }^{2}}{N}\right) +\left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right] \left(\bar{\mi {\mb {x}}}_\mi {N \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot T} - \bar{\bar{\mi {\mb {x}}}}\right)$
$\displaystyle \mr {Cov}\left(D_\mi {i} ^{C}, \tilde{\beta }\right)$	$\displaystyle =-\left(\bar{\mi {\mb {x}}}_\mi {i \cdot } - \bar{\mi {\mb {x}}}_\mi {N \cdot }\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right]$
$\displaystyle \mr {Cov}\left(D_\mi {t} ^{T}, \tilde{\beta }\right)$	$\displaystyle =-\left(\bar{\mi {\mb {x}}}_\mi {\cdot t} - \bar{\mi {\mb {x}}}_\mi {\cdot T}\right)^{} \mr {Var}\left[{\tilde{\beta }}_{s}\right]$
$\displaystyle \mr {Cov}\left(\mr {Intercept}, \tilde{\beta }\right)$	$\displaystyle =-\left(\bar{\mi {\mb {x}}}_\mi {N \cdot } + \bar{\mi {\mb {x}}}_\mi {\cdot T} - \bar{\bar{\mi {\mb {x}}}}\right)^{}\mr {Var}\left[{\tilde{\beta }}_{s}\right]$