The PANEL Procedure

Balanced Panels

Subsections:

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:

\begin{eqnarray*} \mr{Var}\left(D_\emph {i} ^{C}\right) & =& \hat{\sigma }_{\epsilon }^{2} \left(\frac{1}{T} + \frac{1}{N} - \frac{1}{NT} \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] \left(\bar{\mi{\mb{x}}}_\mi {i \cdot } + \bar{\mi{\mb{x}}}_\mi {\cdot T} - \bar{\bar{\mi{\mb{x}}}}\right) \end{eqnarray*}
\[ \mr{Var}\left(D_\mi {t} ^{T}\right) = \frac{2\hat{\sigma }_{\epsilon }^{2}}{\mi{N}} +\left(\bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\mi{\mb{x}}}_\mi {\cdot T}\right)^{'} \mr{Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\mi{\mb{x}}}_\mi {\cdot T}\right) \]
\begin{eqnarray*} \mr{Cov}\left(D_\emph {i} ^{C},D_\emph {j} ^{C}\right) & =& \hat{\sigma }_{\epsilon }^{2} \left(\frac{1}{N} - \frac{1}{NT} \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] \left(\bar{\mi{\mb{x}}}_\mi {j \cdot } + \bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\bar{\mi{\mb{x}}}}\right) \end{eqnarray*}
\[ \mr{Cov}\left(D_\mi {t} ^{T},D_\mi {u} ^{T}\right) = \frac{\hat{\sigma }_{\epsilon }^{2}}{N} + \left(\bar{\mi{\mb{x}}}_\mi {\cdot t} - {\bar{\mi{\mb{x}}}}_{\cdot T}\right)^{'} \mr{Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi{\mb{x}}}_\mi {\cdot u} - \bar{\mi{\mb{x}}}_{\cdot T}\right) \]
\[ \mr{Cov}\left(D_\mi {i} ^{C},D_\mi {t} ^{T}\right) = -\frac{\hat{\sigma }_{\epsilon }^{2}}{N} +\left(\bar{\mi{\mb{x}}}_\mi {i \cdot } + \bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\bar{\mi{\mb{x}}}}\right)^{'}\mr{Var}\left[{\tilde{\beta }}_{s}\right] \left( \bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\mi{\mb{x}}}_{\cdot T}\right) \]
\[ \mr{Cov}\left( D_\mi {i} ^ C,\beta \right) = -\left(\bar{\mi{\mb{x}}}_\mi {i \cdot } + \bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\bar{\mi{\mb{x}}}}\right)^{'}\mr{Var}\left[{\tilde{\beta }}_{s}\right] \]
\[ \mr{Cov}\left( D_\mi {i} ^ T,\beta \right) = -\left(\bar{\mi{\mb{x}}}_\mi {\cdot t} - \bar{\mi{\mb{x}}}_\mi {\cdot T}\right)^{'} \mr{Var}\left[{\tilde{\beta }}_{s}\right] \]

Variance Covariance of Dummy Variables with Intercept

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

\begin{align*} \mr{Var}\left(D_\mi {i} ^{C}\right) & = \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)\\ \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)\\ \mr{Var}\left(\mr{Intercept}\right) & = \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)\\ \mr{Cov}\left(D_\mi {i} ^{C},D_\mi {j} ^{C}\right) & = \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)\\ \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}}}_\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)\\ \mr{Cov}\left(D_\mi {i} ^{C},D_\mi {u} ^{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 {\cdot u} - \bar{\mi{\mb{x}}}_\mi {\cdot T}\right)\\ \mr{Cov}\left(D_\mi {i} ^{C},\mr{Intercept}\right) & =-\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) \\ \mr{Cov}\left(D_\mi {t} ^{T},\mr{Intercept}\right) & =-\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) \\ \mr{Cov}\left(D_\mi {i} ^{C}, \tilde{\beta }\right) & =-\left(\bar{\mi{\mb{x}}}_\mi {i \cdot } - \bar{\mi{\mb{x}}}_\mi {N \cdot }\right)^{'}\mr{Var}\left[{\tilde{\beta }}_{s}\right] \\ \mr{Cov}\left(D_\mi {t} ^{T}, \tilde{\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] \\ \mr{Cov}\left(\mr{Intercept}, \tilde{\beta }\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] \end{align*}