8. The Poolability Test
In order to conduct the poolability test, you need to run group by group OLS regressions and/or time by time OLS regressions. If the null hypothesis is rejected, the panel data are not poolable. In this case, you may consider the random coefficient model and hierarchical regression model.
8.1 Group by Group OLS Regression
In SAS, use the BY statement in the REG procedure. Do not forget to sort the data set in advance.
PROC SORT DATA=masil.airline;
BY airline;
PROC REG DATA=masil.airline;
MODEL cost = output fuel load;
BY airline;
RUN;
}
In Stata, the if qualifier makes it easy to run group by group regressions.
. forvalues i= 1(1)6 { // run group by group regression
display "OLS regression for group " `i'
regress cost output fuel load if airline==`i'
}
OLS
regression for group 1
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 1843.46
Model | 3.41824348 3 1.13941449 Prob > F = 0.0000
Residual | .006798918 11 .000618083 R-squared = 0.9980
-------------+------------------------------ Adj R-squared = 0.9975
Total | 3.4250424 14 .244645886 Root MSE = .02486
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.18318 .0968946 12.21 0.000 .9699164 1.396444
fuel | .3865867 .0181946 21.25 0.000 .3465406 .4266329
load | -2.461629 .4013571 -6.13 0.000 -3.34501 -1.578248
_cons | 10.846 .2972551 36.49 0.000 10.19174 11.50025
------------------------------------------------------------------------------
OLS regression for group 2
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 3129.50
Model | 6.47622084 3 2.15874028 Prob > F = 0.0000
Residual | .007587838 11 .000689803 R-squared = 0.9988
-------------+------------------------------ Adj R-squared = 0.9985
Total | 6.48380868 14 .463129191 Root MSE = .02626
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.459104 .0792856 18.40 0.000 1.284597 1.63361
fuel | .3088958 .0272443 11.34 0.000 .2489315 .36886
load | -2.724785 .2376522 -11.47 0.000 -3.247854 -2.201716
_cons | 11.97243 .4320951 27.71 0.000 11.02139 12.92346
------------------------------------------------------------------------------
OLS regression for group 3
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 608.10
Model | 3.79286673 3 1.26428891 Prob > F = 0.0000
Residual | .022869767 11 .00207907 R-squared = 0.9940
-------------+------------------------------ Adj R-squared = 0.9924
Total | 3.8157365 14 .272552607 Root MSE = .0456
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+----------------------------------------------------------------
output | .7268305 .1554418 4.68 0.001 .3847054 1.068956
fuel | .4515127 .0381103 11.85 0.000 .3676324 .5353929
load | -.7513069 .6105989 -1.23 0.244 -2.095226 .5926122
_cons | 8.699815 .8985786 9.68 0.000 6.722057 10.67757
------------------------------------------------------------------------------
OLS regression for group 4
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 777.86
Model | 7.37252558 3 2.45750853 Prob > F = 0.0000
Residual | .034752343 11 .003159304 R-squared = 0.9953
-------------+------------------------------ Adj R-squared = 0.9940
Total | 7.40727792 14 .52909128 Root MSE = .05621
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .9353749 .0759266 12.32 0.000 .7682616 1.102488
fuel | .4637263 .044347 10.46 0.000 .3661192 .5613333
load | -.7756708 .4707826 -1.65 0.128 -1.811856 .2605148
_cons | 9.164608 .6023241 15.22 0.000 7.838902 10.49031
------------------------------------------------------------------------------
OLS regression for group 5
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 1999.89
Model | 7.08313716 3 2.36104572 Prob > F = 0.0000
Residual | .012986435 11 .001180585 R-squared = 0.9982
-------------+------------------------------ Adj R-squared = 0.9977
Total | 7.09612359 14 .506865971 Root MSE = .03436
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.076299 .0771255 13.96 0.000 .9065471 1.246051
fuel | .2920542 .0434213 6.73 0.000 .1964845 .3876239
load | -1.206847 .3336308 -3.62 0.004 -1.941163 -.4725305
_cons | 11.77079 .7430078 15.84 0.000 10.13544 13.40614
------------------------------------------------------------------------------
OLS regression for group 6
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 2602.49
Model | 11.1173565 3 3.70578551 Prob > F = 0.0000
Residual | .015663323 11 .001423938 R-squared = 0.9986
-------------+------------------------------ Adj R-squared = 0.9982
Total | 11.1330199 14 .795215705 Root MSE = .03774
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .9673393 .0321728 30.07 0.000 .8965275 1.038151
fuel | .3023258 .0308235 9.81 0.000 .2344839 .3701678
load | .1050328 .4767508 0.22 0.830 -.9442886 1.154354
_cons | 10.77381 .4095921 26.30 0.000 9.872309 11.67532
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 1843.46
Model | 3.41824348 3 1.13941449 Prob > F = 0.0000
Residual | .006798918 11 .000618083 R-squared = 0.9980
-------------+------------------------------ Adj R-squared = 0.9975
Total | 3.4250424 14 .244645886 Root MSE = .02486
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.18318 .0968946 12.21 0.000 .9699164 1.396444
fuel | .3865867 .0181946 21.25 0.000 .3465406 .4266329
load | -2.461629 .4013571 -6.13 0.000 -3.34501 -1.578248
_cons | 10.846 .2972551 36.49 0.000 10.19174 11.50025
------------------------------------------------------------------------------
OLS regression for group 2
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 3129.50
Model | 6.47622084 3 2.15874028 Prob > F = 0.0000
Residual | .007587838 11 .000689803 R-squared = 0.9988
-------------+------------------------------ Adj R-squared = 0.9985
Total | 6.48380868 14 .463129191 Root MSE = .02626
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.459104 .0792856 18.40 0.000 1.284597 1.63361
fuel | .3088958 .0272443 11.34 0.000 .2489315 .36886
load | -2.724785 .2376522 -11.47 0.000 -3.247854 -2.201716
_cons | 11.97243 .4320951 27.71 0.000 11.02139 12.92346
------------------------------------------------------------------------------
OLS regression for group 3
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 608.10
Model | 3.79286673 3 1.26428891 Prob > F = 0.0000
Residual | .022869767 11 .00207907 R-squared = 0.9940
-------------+------------------------------ Adj R-squared = 0.9924
Total | 3.8157365 14 .272552607 Root MSE = .0456
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+----------------------------------------------------------------
output | .7268305 .1554418 4.68 0.001 .3847054 1.068956
fuel | .4515127 .0381103 11.85 0.000 .3676324 .5353929
load | -.7513069 .6105989 -1.23 0.244 -2.095226 .5926122
_cons | 8.699815 .8985786 9.68 0.000 6.722057 10.67757
------------------------------------------------------------------------------
OLS regression for group 4
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 777.86
Model | 7.37252558 3 2.45750853 Prob > F = 0.0000
Residual | .034752343 11 .003159304 R-squared = 0.9953
-------------+------------------------------ Adj R-squared = 0.9940
Total | 7.40727792 14 .52909128 Root MSE = .05621
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .9353749 .0759266 12.32 0.000 .7682616 1.102488
fuel | .4637263 .044347 10.46 0.000 .3661192 .5613333
load | -.7756708 .4707826 -1.65 0.128 -1.811856 .2605148
_cons | 9.164608 .6023241 15.22 0.000 7.838902 10.49031
------------------------------------------------------------------------------
OLS regression for group 5
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 1999.89
Model | 7.08313716 3 2.36104572 Prob > F = 0.0000
Residual | .012986435 11 .001180585 R-squared = 0.9982
-------------+------------------------------ Adj R-squared = 0.9977
Total | 7.09612359 14 .506865971 Root MSE = .03436
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | 1.076299 .0771255 13.96 0.000 .9065471 1.246051
fuel | .2920542 .0434213 6.73 0.000 .1964845 .3876239
load | -1.206847 .3336308 -3.62 0.004 -1.941163 -.4725305
_cons | 11.77079 .7430078 15.84 0.000 10.13544 13.40614
------------------------------------------------------------------------------
OLS regression for group 6
Source | SS df MS Number of obs = 15
-------------+------------------------------ F( 3, 11) = 2602.49
Model | 11.1173565 3 3.70578551 Prob > F = 0.0000
Residual | .015663323 11 .001423938 R-squared = 0.9986
-------------+------------------------------ Adj R-squared = 0.9982
Total | 11.1330199 14 .795215705 Root MSE = .03774
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .9673393 .0321728 30.07 0.000 .8965275 1.038151
fuel | .3023258 .0308235 9.81 0.000 .2344839 .3701678
load | .1050328 .4767508 0.22 0.830 -.9442886 1.154354
_cons | 10.77381 .4095921 26.30 0.000 9.872309 11.67532
------------------------------------------------------------------------------
8.2 Poolability Test across Groups
The null hypothesis of the poolability test across groups is that all group parameters are equal to corresponding pooled parameters. The e’e is 1.3354, the SSE of the pooled OLS regression. The sum of SSEi is .1007 = .0068 + .0076 + .0229 + .0348 + .0130 + .0157.
Thus, the F statistic is
The large 40.4812 rejects the null hypothesis of poolability (p< .0000). We conclude that the panel data are not poolable with respect to group.
8.3 Poolability Test over Time
The null hypothesis of the poolability test over time is that all time parameters are equal to corresponding pooled parameters. The sum of SSEt is computed from the 15 time by time regression.
. di .044807673 + .023093978 + .016506613 + .012170358 + .014104542 + ///
.000469826 + .063648817 + .085430285 + .049329439 + .077112957 + ///
.029913538 + .087240016 + .143348297 + .066075346 + .037256216
.000469826 + .063648817 + .085430285 + .049329439 + .077112957 + ///
.029913538 + .087240016 + .143348297 + .066075346 + .037256216
.7505079
The F statistic is
The small F statistic does not reject the null hypothesis in favor of poolable panel data with respect to time (p<.9991).
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