6. The Fixed Group and Time Effect Model
The two-way fixed model considers both group and time effects. This model thus needs two sets of group and time dummy variables. LSDV2 and the between effect model are not valid in this model.
6.1 Least Squares Dummy Variable Models
There are four approaches to avoid the perfect multicollinearity or the dummy variable trap. You may not suppress the intercept under any circumstances.
- Drop one cross-section and one time-series dummy variables
- Drop one cross-section dummy and impose a restriction on the time-series dummies
- Drop one time-series dummy and impose a restriction on the cross-section dummies
- Include all dummy variables and impose two restrictions on the cross-section and time-series dummies
Let us first run LSDV1 using STATA.
. regress cost g1-g5 t1-t14 output fuel load
Source |
SS df
MS
Number of obs = 90
-------------+------------------------------ F( 22, 67) = 1960.82
Model | 113.864044 22 5.17563838 Prob > F = 0.0000
Residual | .176848775 67 .002639534 R-squared = 0.9984
-------------+------------------------------ Adj R-squared = 0.9979
Total | 114.040893 89 1.28135835 Root MSE = .05138
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1742825 .0861201 2.02 0.047 .0023861 .346179
g2 | .1114508 .0779551 1.43 0.157 -.0441482 .2670499
g3 | -.143511 .0518934 -2.77 0.007 -.2470907 -.0399313
g4 | .1802087 .0321443 5.61 0.000 .1160484 .2443691
g5 | -.0466942 .0224688 -2.08 0.042 -.0915422 -.0018463
t1 | -.6931382 .3378385 -2.05 0.044 -1.367467 -.0188098
t2 | -.6384366 .3320802 -1.92 0.059 -1.301271 .0243983
t3 | -.5958031 .3294473 -1.81 0.075 -1.253383 .0617764
t4 | -.5421537 .3189139 -1.70 0.094 -1.178708 .0944011
t5 | -.4730429 .2319459 -2.04 0.045 -.9360088 -.0100769
t6 | -.4272042 .18844 -2.27 0.027 -.8033319 -.0510764
t7 | -.3959783 .1732969 -2.28 0.025 -.7418804 -.0500762
t8 | -.3398463 .1501062 -2.26 0.027 -.6394596 -.040233
t9 | -.2718933 .1348175 -2.02 0.048 -.5409901 -.0027964
t10 | -.2273857 .0763495 -2.98 0.004 -.37978 -.0749914
t11 | -.1118032 .0319005 -3.50 0.001 -.175477 -.0481295
t12 | -.033641 .0429008 -0.78 0.436 -.1192713 .0519893
t13 | -.0177346 .0362554 -0.49 0.626 -.0901007 .0546315
t14 | -.0186451 .030508 -0.61 0.543 -.0795393 .042249
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.94004 2.218231 5.83 0.000 8.512434 17.36765
------------------------------------------------------------------------------
-------------+------------------------------ F( 22, 67) = 1960.82
Model | 113.864044 22 5.17563838 Prob > F = 0.0000
Residual | .176848775 67 .002639534 R-squared = 0.9984
-------------+------------------------------ Adj R-squared = 0.9979
Total | 114.040893 89 1.28135835 Root MSE = .05138
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1742825 .0861201 2.02 0.047 .0023861 .346179
g2 | .1114508 .0779551 1.43 0.157 -.0441482 .2670499
g3 | -.143511 .0518934 -2.77 0.007 -.2470907 -.0399313
g4 | .1802087 .0321443 5.61 0.000 .1160484 .2443691
g5 | -.0466942 .0224688 -2.08 0.042 -.0915422 -.0018463
t1 | -.6931382 .3378385 -2.05 0.044 -1.367467 -.0188098
t2 | -.6384366 .3320802 -1.92 0.059 -1.301271 .0243983
t3 | -.5958031 .3294473 -1.81 0.075 -1.253383 .0617764
t4 | -.5421537 .3189139 -1.70 0.094 -1.178708 .0944011
t5 | -.4730429 .2319459 -2.04 0.045 -.9360088 -.0100769
t6 | -.4272042 .18844 -2.27 0.027 -.8033319 -.0510764
t7 | -.3959783 .1732969 -2.28 0.025 -.7418804 -.0500762
t8 | -.3398463 .1501062 -2.26 0.027 -.6394596 -.040233
t9 | -.2718933 .1348175 -2.02 0.048 -.5409901 -.0027964
t10 | -.2273857 .0763495 -2.98 0.004 -.37978 -.0749914
t11 | -.1118032 .0319005 -3.50 0.001 -.175477 -.0481295
t12 | -.033641 .0429008 -0.78 0.436 -.1192713 .0519893
t13 | -.0177346 .0362554 -0.49 0.626 -.0901007 .0546315
t14 | -.0186451 .030508 -0.61 0.543 -.0795393 .042249
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.94004 2.218231 5.83 0.000 8.512434 17.36765
------------------------------------------------------------------------------
The following is the corresponding SAS REG procedure (outputs are skipped).
PROC REG DATA=masil.airline;
MODEL cost = g1-g5 t1-t14 output fuel load;
RUN;
The LIMDEP example is skipped here, since many dummy variables need to be listed in the Regress$ command.
6.3 LSDV1 + LSDV3: Dropping a Dummy and Imposing a Restriction
In the second approach, you may drop either one group dummy or one time dummy. The following drops one time dummy, includes all group dummies, and imposes a restriction on group dummies.
PROC REG DATA=masil.airline;
MODEL cost = g1-g6 t1-t14 output fuel load;
RESTRICT g1 + g2 + g3 + g4 + g5 + g6 = 0;
RUN;
The REG Procedure
Model: MODEL1
Dependent Variable: cost
NOTE: Restrictions have been applied to parameter estimates.
Number of Observations Read 90
Number of Observations Used 90
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 22 113.86404 5.17564 1960.82 <.0001
Error 67 0.17685 0.00264
Corrected Total 89 114.04089
Root MSE 0.05138 R-Square 0.9984
Dependent Mean 13.36561 Adj R-Sq 0.9979
Coeff Var 0.38439
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 12.98600 2.22540 5.84 <.0001
g1 1 0.12833 0.04601 2.79 0.0069
g2 1 0.06549 0.03897 1.68 0.0975
g3 1 -0.18947 0.01561 -12.14 <.0001
g4 1 0.13425 0.01832 7.33 <.0001
g5 1 -0.09265 0.03731 -2.48 0.0155
g6 1 -0.04596 0.04161 -1.10 0.2733
t1 1 -0.69314 0.33784 -2.05 0.0441
t2 1 -0.63844 0.33208 -1.92 0.0588
t3 1 -0.59580 0.32945 -1.81 0.0750
t4 1 -0.54215 0.31891 -1.70 0.0938
t5 1 -0.47304 0.23195 -2.04 0.0454
t6 1 -0.42720 0.18844 -2.27 0.0266
t7 1 -0.39598 0.17330 -2.28 0.0255
t8 1 -0.33985 0.15011 -2.26 0.0268
t9 1 -0.27189 0.13482 -2.02 0.0477
t10 1 -0.22739 0.07635 -2.98 0.0040
t11 1 -0.11180 0.03190 -3.50 0.0008
t12 1 -0.03364 0.04290 -0.78 0.4357
t13 1 -0.01773 0.03626 -0.49 0.6263
t14 1 -0.01865 0.03051 -0.61 0.5432
output 1 0.81725 0.03185 25.66 <.0001
fuel 1 0.16861 0.16348 1.03 0.3061
load 1 -0.88281 0.26174 -3.37 0.0012
RESTRICT -1 -1.9387E-16 . . .
* Probability computed using beta distribution.
Model: MODEL1
Dependent Variable: cost
NOTE: Restrictions have been applied to parameter estimates.
Number of Observations Read 90
Number of Observations Used 90
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 22 113.86404 5.17564 1960.82 <.0001
Error 67 0.17685 0.00264
Corrected Total 89 114.04089
Root MSE 0.05138 R-Square 0.9984
Dependent Mean 13.36561 Adj R-Sq 0.9979
Coeff Var 0.38439
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 12.98600 2.22540 5.84 <.0001
g1 1 0.12833 0.04601 2.79 0.0069
g2 1 0.06549 0.03897 1.68 0.0975
g3 1 -0.18947 0.01561 -12.14 <.0001
g4 1 0.13425 0.01832 7.33 <.0001
g5 1 -0.09265 0.03731 -2.48 0.0155
g6 1 -0.04596 0.04161 -1.10 0.2733
t1 1 -0.69314 0.33784 -2.05 0.0441
t2 1 -0.63844 0.33208 -1.92 0.0588
t3 1 -0.59580 0.32945 -1.81 0.0750
t4 1 -0.54215 0.31891 -1.70 0.0938
t5 1 -0.47304 0.23195 -2.04 0.0454
t6 1 -0.42720 0.18844 -2.27 0.0266
t7 1 -0.39598 0.17330 -2.28 0.0255
t8 1 -0.33985 0.15011 -2.26 0.0268
t9 1 -0.27189 0.13482 -2.02 0.0477
t10 1 -0.22739 0.07635 -2.98 0.0040
t11 1 -0.11180 0.03190 -3.50 0.0008
t12 1 -0.03364 0.04290 -0.78 0.4357
t13 1 -0.01773 0.03626 -0.49 0.6263
t14 1 -0.01865 0.03051 -0.61 0.5432
output 1 0.81725 0.03185 25.66 <.0001
fuel 1 0.16861 0.16348 1.03 0.3061
load 1 -0.88281 0.26174 -3.37 0.0012
RESTRICT -1 -1.9387E-16 . . .
* Probability computed using beta distribution.
Alternatively, you may run the STATA .cnsreg command with the second constraint (output is skipped).
. cnsreg cost g1-g6 t1-t14 output fuel load, constraint(2)
The following drops one group dummy and imposes a restriction on time dummies.
. cnsreg cost g1-g5 t1-t15 output fuel load, constraint(3)
Constrained
linear
regression
Number of obs = 90
F( 22, 67) = 1960.82
Prob > F = 0.0000
Root MSE = .05138
( 1) t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9 + t10 + t11 + t12 + t13 + t14 + t15 = 0
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1742825 .0861201 2.02 0.047 .0023861 .346179
g2 | .1114508 .0779551 1.43 0.157 -.0441482 .2670499
g3 | -.143511 .0518934 -2.77 0.007 -.2470907 -.0399313
g4 | .1802087 .0321443 5.61 0.000 .1160484 .2443691
g5 | -.0466942 .0224688 -2.08 0.042 -.0915422 -.0018463
t1 | -.3740245 .191872 -1.95 0.055 -.7570026 .0089536
t2 | -.3193228 .1860877 -1.72 0.091 -.6907554 .0521097
t3 | -.2766893 .1833501 -1.51 0.136 -.6426576 .0892789
t4 | -.2230399 .1729671 -1.29 0.202 -.5682837 .1222038
t5 | -.1539291 .0864404 -1.78 0.079 -.3264649 .0186066
t6 | -.1080904 .0448591 -2.41 0.019 -.1976296 -.0185513
t7 | -.0768646 .0319336 -2.41 0.019 -.1406043 -.0131248
t8 | -.0207326 .0204506 -1.01 0.314 -.061552 .0200869
t9 | .0472205 .0290822 1.62 0.109 -.0108278 .1052688
t10 | .0917281 .0811525 1.13 0.262 -.0702531 .2537092
t11 | .2073105 .1491443 1.39 0.169 -.0903829 .5050039
t12 | .2854727 .1756365 1.63 0.109 -.0650993 .6360447
t13 | .3013791 .1660294 1.82 0.074 -.030017 .6327752
t14 | .3004686 .1536212 1.96 0.055 -.0061606 .6070978
t15 | .3191137 .1474883 2.16 0.034 .0247259 .6135015
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.62093 2.074302 6.08 0.000 8.480603 16.76125
------------------------------------------------------------------------------
F( 22, 67) = 1960.82
Prob > F = 0.0000
Root MSE = .05138
( 1) t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9 + t10 + t11 + t12 + t13 + t14 + t15 = 0
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1742825 .0861201 2.02 0.047 .0023861 .346179
g2 | .1114508 .0779551 1.43 0.157 -.0441482 .2670499
g3 | -.143511 .0518934 -2.77 0.007 -.2470907 -.0399313
g4 | .1802087 .0321443 5.61 0.000 .1160484 .2443691
g5 | -.0466942 .0224688 -2.08 0.042 -.0915422 -.0018463
t1 | -.3740245 .191872 -1.95 0.055 -.7570026 .0089536
t2 | -.3193228 .1860877 -1.72 0.091 -.6907554 .0521097
t3 | -.2766893 .1833501 -1.51 0.136 -.6426576 .0892789
t4 | -.2230399 .1729671 -1.29 0.202 -.5682837 .1222038
t5 | -.1539291 .0864404 -1.78 0.079 -.3264649 .0186066
t6 | -.1080904 .0448591 -2.41 0.019 -.1976296 -.0185513
t7 | -.0768646 .0319336 -2.41 0.019 -.1406043 -.0131248
t8 | -.0207326 .0204506 -1.01 0.314 -.061552 .0200869
t9 | .0472205 .0290822 1.62 0.109 -.0108278 .1052688
t10 | .0917281 .0811525 1.13 0.262 -.0702531 .2537092
t11 | .2073105 .1491443 1.39 0.169 -.0903829 .5050039
t12 | .2854727 .1756365 1.63 0.109 -.0650993 .6360447
t13 | .3013791 .1660294 1.82 0.074 -.030017 .6327752
t14 | .3004686 .1536212 1.96 0.055 -.0061606 .6070978
t15 | .3191137 .1474883 2.16 0.034 .0247259 .6135015
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.62093 2.074302 6.08 0.000 8.480603 16.76125
------------------------------------------------------------------------------
You may run the following SAS REG procedure to get the same result (output is skipped).
PROC REG DATA=masil.airline; /* LSDV3 */
MODEL cost = g1-g5 t1-t15 output fuel load;
RESTRICT t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15=0;
RUN;
6.4 LSDV3 with Two Restrictions
The third approach includes all group and time dummies and imposes two restrictions on group and time dummies.
. cnsreg cost g1-g6 t1-t15 output fuel load, constraint(2 3)
Constrained
linear
regression
Number of obs = 90
F( 22, 67) = 1960.82
Prob > F = 0.0000
Root MSE = .05138
( 1) g1 + g2 + g3 + g4 + g5 + g6 = 0
( 2) t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9 + t10 + t11 + t12 + t13 + t14 + t15 = 0
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1283264 .0460126 2.79 0.007 .0364849 .2201679
g2 | .0654947 .0389685 1.68 0.097 -.0122867 .1432761
g3 | -.1894671 .0156096 -12.14 0.000 -.220624 -.1583102
g4 | .1342526 .0183163 7.33 0.000 .097693 .1708121
g5 | -.0926504 .0373085 -2.48 0.016 -.1671184 -.0181824
g6 | -.0459561 .0416069 -1.10 0.273 -.1290038 .0370916
t1 | -.3740245 .191872 -1.95 0.055 -.7570026 .0089536
t2 | -.3193228 .1860877 -1.72 0.091 -.6907554 .0521097
t3 | -.2766893 .1833501 -1.51 0.136 -.6426576 .0892789
t4 | -.2230399 .1729671 -1.29 0.202 -.5682837 .1222038
t5 | -.1539291 .0864404 -1.78 0.079 -.3264649 .0186066
t6 | -.1080904 .0448591 -2.41 0.019 -.1976296 -.0185513
t7 | -.0768646 .0319336 -2.41 0.019 -.1406043 -.0131248
t8 | -.0207326 .0204506 -1.01 0.314 -.061552 .0200869
t9 | .0472205 .0290822 1.62 0.109 -.0108278 .1052688
t10 | .0917281 .0811525 1.13 0.262 -.0702531 .2537092
t11 | .2073105 .1491443 1.39 0.169 -.0903829 .5050039
t12 | .2854727 .1756365 1.63 0.109 -.0650993 .6360447
t13 | .3013791 .1660294 1.82 0.074 -.030017 .6327752
t14 | .3004686 .1536212 1.96 0.055 -.0061606 .6070978
t15 | .3191137 .1474883 2.16 0.034 .0247259 .6135015
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.66688 2.081068 6.09 0.000 8.513054 16.82071
------------------------------------------------------------------------------
F( 22, 67) = 1960.82
Prob > F = 0.0000
Root MSE = .05138
( 1) g1 + g2 + g3 + g4 + g5 + g6 = 0
( 2) t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9 + t10 + t11 + t12 + t13 + t14 + t15 = 0
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
g1 | .1283264 .0460126 2.79 0.007 .0364849 .2201679
g2 | .0654947 .0389685 1.68 0.097 -.0122867 .1432761
g3 | -.1894671 .0156096 -12.14 0.000 -.220624 -.1583102
g4 | .1342526 .0183163 7.33 0.000 .097693 .1708121
g5 | -.0926504 .0373085 -2.48 0.016 -.1671184 -.0181824
g6 | -.0459561 .0416069 -1.10 0.273 -.1290038 .0370916
t1 | -.3740245 .191872 -1.95 0.055 -.7570026 .0089536
t2 | -.3193228 .1860877 -1.72 0.091 -.6907554 .0521097
t3 | -.2766893 .1833501 -1.51 0.136 -.6426576 .0892789
t4 | -.2230399 .1729671 -1.29 0.202 -.5682837 .1222038
t5 | -.1539291 .0864404 -1.78 0.079 -.3264649 .0186066
t6 | -.1080904 .0448591 -2.41 0.019 -.1976296 -.0185513
t7 | -.0768646 .0319336 -2.41 0.019 -.1406043 -.0131248
t8 | -.0207326 .0204506 -1.01 0.314 -.061552 .0200869
t9 | .0472205 .0290822 1.62 0.109 -.0108278 .1052688
t10 | .0917281 .0811525 1.13 0.262 -.0702531 .2537092
t11 | .2073105 .1491443 1.39 0.169 -.0903829 .5050039
t12 | .2854727 .1756365 1.63 0.109 -.0650993 .6360447
t13 | .3013791 .1660294 1.82 0.074 -.030017 .6327752
t14 | .3004686 .1536212 1.96 0.055 -.0061606 .6070978
t15 | .3191137 .1474883 2.16 0.034 .0247259 .6135015
output | .8172487 .031851 25.66 0.000 .7536739 .8808235
fuel | .16861 .163478 1.03 0.306 -.1576935 .4949135
load | -.8828142 .2617373 -3.37 0.001 -1.405244 -.3603843
_cons | 12.66688 2.081068 6.09 0.000 8.513054 16.82071
------------------------------------------------------------------------------
The following SAS REG procedure gives you the same result (output is skipped).
PROC REG DATA=masil.airline;
MODEL cost = g1-g6 t1-t15 output fuel load;
RESTRICT g1 + g2 + g3 + g4 + g5 + g6 = 0;
RESTRICT t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15=0;
RUN;
6.5 Two-way Within Effect Model
The two-way within group and time effect model requires a transformation of the data set. The following commands do this task.
. gen w_cost = cost - gm_cost - tm_cost + m_cost
. gen w_output = output - gm_output - tm_output + m_output
. gen w_fuel = fuel - gm_fuel - tm_fuel + m_fuel
. gen w_load = load - gm_load - tm_load + m_load
. tabstat cost output fuel load, stat(mean)
stats | cost
output fuel load
---------+----------------------------------------
mean | 13.36561 -1.174309 12.77036 .5604602
--------------------------------------------------
---------+----------------------------------------
mean | 13.36561 -1.174309 12.77036 .5604602
--------------------------------------------------
Now, run the OLS with the transformed variables. Do not forget to suppress the intercept.
. regress w_cost w_output w_fuel w_load, noc // within effect
Source |
SS df
MS
Number of obs = 90
-------------+------------------------------ F( 3, 87) = 307.86
Model | 1.87739643 3 .625798811 Prob > F = 0.0000
Residual | .176848774 87 .002032745 R-squared = 0.9139
-------------+------------------------------ Adj R-squared = 0.9109
Total | 2.05424521 90 .022824947 Root MSE = .04509
------------------------------------------------------------------------------
w_cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w_output | .8172487 .0279512 29.24 0.000 .7616927 .8728048
w_fuel | .16861 .1434621 1.18 0.243 -.1165364 .4537565
w_load | -.8828142 .2296907 -3.84 0.000 -1.339349 -.426279
------------------------------------------------------------------------------
-------------+------------------------------ F( 3, 87) = 307.86
Model | 1.87739643 3 .625798811 Prob > F = 0.0000
Residual | .176848774 87 .002032745 R-squared = 0.9139
-------------+------------------------------ Adj R-squared = 0.9109
Total | 2.05424521 90 .022824947 Root MSE = .04509
------------------------------------------------------------------------------
w_cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
w_output | .8172487 .0279512 29.24 0.000 .7616927 .8728048
w_fuel | .16861 .1434621 1.18 0.243 -.1165364 .4537565
w_load | -.8828142 .2296907 -3.84 0.000 -1.339349 -.426279
------------------------------------------------------------------------------
Note again that R2, MSE, standard errors, and DF of error are not correct. The dummy variable coefficients need to be computed. The standard errors also need to be adjusted; for instance, the standard error of the load factor is .2617=.2297*sqrt(87/67).
6.6 Using the TSCSREG and PANEL Procedures
The SAS TSCSREG and PANEL procedures have the /FIXTWO option to fit the two-way fixed effect model.
PROC TSCSREG DATA=masil.airline;
ID airline year;
MODEL cost = output fuel load /FIXTWO;
RUN;
The TSCSREG Procedure
Dependent Variable: cost
Model Description
Estimation Method FixTwo
Number of Cross Sections 6
Time Series Length 15
Fit Statistics
SSE 0.1768 DFE 67
MSE 0.0026 Root MSE 0.0514
R-Square 0.9984
F Test for No Fixed Effects
Num DF Den DF F Value Pr > F
19 67 23.10 <.0001
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
CS1 1 0.174283 0.0861 2.02 0.0470 Cross Sectional
Effect 1
CS2 1 0.111451 0.0780 1.43 0.1575 Cross Sectional
Effect 2
CS3 1 -0.14351 0.0519 -2.77 0.0073 Cross Sectional
Effect 3
CS4 1 0.180209 0.0321 5.61 <.0001 Cross Sectional
Effect 4
CS5 1 -0.04669 0.0225 -2.08 0.0415 Cross Sectional
Effect 5
TS1 1 -0.69314 0.3378 -2.05 0.0441 Time Series
Effect 1
TS2 1 -0.63844 0.3321 -1.92 0.0588 Time Series
Effect 2
TS3 1 -0.5958 0.3294 -1.81 0.0750 Time Series
Effect 3
TS4 1 -0.54215 0.3189 -1.70 0.0938 Time Series
Effect 4
TS5 1 -0.47304 0.2319 -2.04 0.0454 Time Series
Effect 5
TS6 1 -0.4272 0.1884 -2.27 0.0266 Time Series
Effect 6
TS7 1 -0.39598 0.1733 -2.28 0.0255 Time Series
Effect 7
TS8 1 -0.33985 0.1501 -2.26 0.0268 Time Series
Effect 8
TS9 1 -0.27189 0.1348 -2.02 0.0477 Time Series
Effect 9
TS10 1 -0.22739 0.0763 -2.98 0.0040 Time Series
Effect 10
TS11 1 -0.1118 0.0319 -3.50 0.0008 Time Series
Effect 11
TS12 1 -0.03364 0.0429 -0.78 0.4357 Time Series
Effect 12
TS13 1 -0.01773 0.0363 -0.49 0.6263 Time Series
Effect 13
TS14 1 -0.01865 0.0305 -0.61 0.5432 Time Series
Effect 14
Intercept 1 12.94004 2.2182 5.83 <.0001 Intercept
output 1 0.817249 0.0319 25.66 <.0001
fuel 1 0.16861 0.1635 1.03 0.3061
load 1 -0.88281 0.2617 -3.37 0.0012
Dependent Variable: cost
Model Description
Estimation Method FixTwo
Number of Cross Sections 6
Time Series Length 15
Fit Statistics
SSE 0.1768 DFE 67
MSE 0.0026 Root MSE 0.0514
R-Square 0.9984
F Test for No Fixed Effects
Num DF Den DF F Value Pr > F
19 67 23.10 <.0001
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
CS1 1 0.174283 0.0861 2.02 0.0470 Cross Sectional
Effect 1
CS2 1 0.111451 0.0780 1.43 0.1575 Cross Sectional
Effect 2
CS3 1 -0.14351 0.0519 -2.77 0.0073 Cross Sectional
Effect 3
CS4 1 0.180209 0.0321 5.61 <.0001 Cross Sectional
Effect 4
CS5 1 -0.04669 0.0225 -2.08 0.0415 Cross Sectional
Effect 5
TS1 1 -0.69314 0.3378 -2.05 0.0441 Time Series
Effect 1
TS2 1 -0.63844 0.3321 -1.92 0.0588 Time Series
Effect 2
TS3 1 -0.5958 0.3294 -1.81 0.0750 Time Series
Effect 3
TS4 1 -0.54215 0.3189 -1.70 0.0938 Time Series
Effect 4
TS5 1 -0.47304 0.2319 -2.04 0.0454 Time Series
Effect 5
TS6 1 -0.4272 0.1884 -2.27 0.0266 Time Series
Effect 6
TS7 1 -0.39598 0.1733 -2.28 0.0255 Time Series
Effect 7
TS8 1 -0.33985 0.1501 -2.26 0.0268 Time Series
Effect 8
TS9 1 -0.27189 0.1348 -2.02 0.0477 Time Series
Effect 9
TS10 1 -0.22739 0.0763 -2.98 0.0040 Time Series
Effect 10
TS11 1 -0.1118 0.0319 -3.50 0.0008 Time Series
Effect 11
TS12 1 -0.03364 0.0429 -0.78 0.4357 Time Series
Effect 12
TS13 1 -0.01773 0.0363 -0.49 0.6263 Time Series
Effect 13
TS14 1 -0.01865 0.0305 -0.61 0.5432 Time Series
Effect 14
Intercept 1 12.94004 2.2182 5.83 <.0001 Intercept
output 1 0.817249 0.0319 25.66 <.0001
fuel 1 0.16861 0.1635 1.03 0.3061
load 1 -0.88281 0.2617 -3.37 0.0012
The STATA .xtreg command does not fit the two-way fixed or random effect model. The following LIMDEP command fits the two-way fixed model. Note that this command has Str$ and Period$ specifications to specify stratification and time variables. This command presents the pooled model and one-way group effect model as well, but reports the incorrect intercept in the two-way fixed model, 12.667 (2.081).
REGRESS;Lhs=COST;Rhs=ONE,OUTPUT,FUEL,LOAD;Panel;Str=AIRLINE;Period=YEAR;
Fixed$
6.7 Testing Fixed Group and Time Effects
The null hypothesis is that parameters of group and time dummies are zero. The F test compares the pooled regression and two-way group and time effect model. The F statistic of 23.1085 rejects the null hypothesis at the .01 significance level (p<.0000).
The SAS TSCSREG and PANEL procedures conduct the F-test for the group and time effects. You may also run the following SAS REG procedure and .regress command to perform the same test.
PROC REG DATA=masil.airline;
MODEL cost = g1-g5 t1-t14 output fuel load;
TEST g1=g2=g3=g4=g5=t1=t2=t3=t4=t5=t6=t7=t8=t9=t10=t11=t12=t13=t14=0;
RUN;
. quietly regress cost g1-g5 t1-t14 output fuel load
. test g1 g2 g3 g4 g5 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14
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