Balanced Panels :: SAS/ETS(R) 13.1 User's Guide: High-Performance Procedures

Balanced Panels

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

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

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

where the symbols are as follows:

$\mi {y_\mi {it}}$ 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 following calculations of cross-sectional dummy variables, time dummy variables, and intercepts are similar to how they are calculated 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:

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

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

Use the superscript C and T to denote the cross-sectional dummy variables and time dummy variables, respectively. Under the NOINT option, the following equations produce 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 covariance matrix of ${\tilde{\beta }}_{s}$ is given by

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

For information about the covariance matrix that is related to dummy variables, see the section Two-Way Fixed-Effects Model in SAS/ETS User's Guide.

The HPPANEL Procedure (Experimental)

Balanced Panels