5. The Fixed Time Effect Model
The fixed time effect model investigates how time affects the intercept using time dummy variables. The logic and method are the same as those of the fixed group effect model.
5.1 Least Squares Dummy Variable Models
The least squares dummy variable (LSDV) model produces fifteen regression equations. This section does not present all outputs, but one or two for each LSDV approach.
Time01: cost = 20.496 + .868*output - .484*fuel -1.954*load
Time02: cost = 20.578 + .868*output - .484*fuel -1.954*load
Time03: cost = 20.656 + .868*output - .484*fuel -1.954*load
Time04: cost = 20.741 + .868*output - .484*fuel -1.954*load
Time05: cost = 21.200 + .868*output - .484*fuel -1.954*load
Time06: cost = 21.412 + .868*output - .484*fuel -1.954*load
Time07: cost = 21.503 + .868*output - .484*fuel -1.954*load
Time08: cost = 21.654 + .868*output - .484*fuel -1.954*load
Time09: cost = 21.830 + .868*output - .484*fuel -1.954*load
Time10: cost = 22.114 + .868*output - .484*fuel -1.954*load
Time11: cost = 22.465 + .868*output - .484*fuel -1.954*load
Time12: cost = 22.651 + .868*output - .484*fuel -1.954*load
Time13: cost = 22.617 + .868*output - .484*fuel -1.954*load
Time14: cost = 22.552 + .868*output - .484*fuel -1.954*load
Time15: cost = 22.537 + .868*output - .484*fuel -1.954*load
Time02: cost = 20.578 + .868*output - .484*fuel -1.954*load
Time03: cost = 20.656 + .868*output - .484*fuel -1.954*load
Time04: cost = 20.741 + .868*output - .484*fuel -1.954*load
Time05: cost = 21.200 + .868*output - .484*fuel -1.954*load
Time06: cost = 21.412 + .868*output - .484*fuel -1.954*load
Time07: cost = 21.503 + .868*output - .484*fuel -1.954*load
Time08: cost = 21.654 + .868*output - .484*fuel -1.954*load
Time09: cost = 21.830 + .868*output - .484*fuel -1.954*load
Time10: cost = 22.114 + .868*output - .484*fuel -1.954*load
Time11: cost = 22.465 + .868*output - .484*fuel -1.954*load
Time12: cost = 22.651 + .868*output - .484*fuel -1.954*load
Time13: cost = 22.617 + .868*output - .484*fuel -1.954*load
Time14: cost = 22.552 + .868*output - .484*fuel -1.954*load
Time15: cost = 22.537 + .868*output - .484*fuel -1.954*load
Let us begin with the SAS REG procedure. The test statement examines fixed time effects.
PROC REG DATA=masil.airline;
MODEL cost = t1-t14 output fuel load;
RUN;
The REG Procedure
Model: MODEL1
Dependent Variable: cost
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 17 112.95270 6.64428 439.62 <.0001
Error 72 1.08819 0.01511
Corrected Total 89 114.04089
Root MSE 0.12294 R-Square 0.9905
Dependent Mean 13.36561 Adj R-Sq 0.9882
Coeff Var 0.91981
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 22.53677 4.94053 4.56 <.0001
t1 1 -2.04096 0.73469 -2.78 0.0070
t2 1 -1.95873 0.72275 -2.71 0.0084
t3 1 -1.88103 0.72036 -2.61 0.0110
t4 1 -1.79601 0.69882 -2.57 0.0122
t5 1 -1.33693 0.50604 -2.64 0.0101
t6 1 -1.12514 0.40862 -2.75 0.0075
t7 1 -1.03341 0.37642 -2.75 0.0076
t8 1 -0.88274 0.32601 -2.71 0.0085
t9 1 -0.70719 0.29470 -2.40 0.0190
t10 1 -0.42296 0.16679 -2.54 0.0134
t11 1 -0.07144 0.07176 -1.00 0.3228
t12 1 0.11457 0.09841 1.16 0.2482
t13 1 0.07979 0.08442 0.95 0.3477
t14 1 0.01546 0.07264 0.21 0.8320
output 1 0.86773 0.01541 56.32 <.0001
fuel 1 -0.48448 0.36411 -1.33 0.1875
load 1 -1.95440 0.44238 -4.42 <.0001
Model: MODEL1
Dependent Variable: cost
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 17 112.95270 6.64428 439.62 <.0001
Error 72 1.08819 0.01511
Corrected Total 89 114.04089
Root MSE 0.12294 R-Square 0.9905
Dependent Mean 13.36561 Adj R-Sq 0.9882
Coeff Var 0.91981
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 22.53677 4.94053 4.56 <.0001
t1 1 -2.04096 0.73469 -2.78 0.0070
t2 1 -1.95873 0.72275 -2.71 0.0084
t3 1 -1.88103 0.72036 -2.61 0.0110
t4 1 -1.79601 0.69882 -2.57 0.0122
t5 1 -1.33693 0.50604 -2.64 0.0101
t6 1 -1.12514 0.40862 -2.75 0.0075
t7 1 -1.03341 0.37642 -2.75 0.0076
t8 1 -0.88274 0.32601 -2.71 0.0085
t9 1 -0.70719 0.29470 -2.40 0.0190
t10 1 -0.42296 0.16679 -2.54 0.0134
t11 1 -0.07144 0.07176 -1.00 0.3228
t12 1 0.11457 0.09841 1.16 0.2482
t13 1 0.07979 0.08442 0.95 0.3477
t14 1 0.01546 0.07264 0.21 0.8320
output 1 0.86773 0.01541 56.32 <.0001
fuel 1 -0.48448 0.36411 -1.33 0.1875
load 1 -1.95440 0.44238 -4.42 <.0001
The following are the corresponding STATA and LIMDEP commands for LSDV1 (outputs are skipped).
. regress cost t1-t14 output fuel load
REGRESS;Lhs=COST;Rhs=ONE,T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,
OUTPUT,FUEL,LOAD$
5.1.2 LSDV2 without the Intercept
Let us use LIMDEP to fit LSDV2 because it reports correct (although slightly different) F and R2 statistics.
--> REGRESS;Lhs=COST;Rhs=T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,
T15,OUTPUT,FUEL,LOAD$
+-----------------------------------------------------------------------+
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560929 , S.D.= 1.131971002 |
| Model size: Observations = 90, Parameters = 18, Deg.Fr.= 72 |
| Residuals: Sum of squares= 1.088190223 , Std.Dev.= .12294 |
| Fit: R-squared= .990458, Adjusted R-squared = .98820 |
| Model test: F[ 17, 72] = 439.62, Prob value = .00000 |
| Diagnostic: Log-L = 70.9837, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.010, Akaike Info. Crt.= -1.177 |
| Model does not contain ONE. R-squared and F can be negative! |
| Autocorrel: Durbin-Watson Statistic = 2.93900, Rho = -.46950 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
T1 20.49580478 4.2095283 4.869 .0000 .66666667E-01
T2 20.57803885 4.2215262 4.875 .0000 .66666667E-01
T3 20.65573100 4.2241771 4.890 .0000 .66666667E-01
T4 20.74075857 4.2457497 4.885 .0000 .66666667E-01
T5 21.19983202 4.4403312 4.774 .0000 .66666667E-01
T6 21.41162082 4.5386212 4.718 .0000 .66666667E-01
T7 21.50335085 4.5713968 4.704 .0000 .66666667E-01
T8 21.65402827 4.6228858 4.684 .0000 .66666667E-01
T9 21.82957108 4.6569062 4.688 .0000 .66666667E-01
T10 22.11380260 4.7926483 4.614 .0000 .66666667E-01
T11 22.46532734 4.9499089 4.539 .0000 .66666667E-01
T12 22.65133704 5.0085924 4.522 .0000 .66666667E-01
T13 22.61655508 4.9861391 4.536 .0000 .66666667E-01
T14 22.55222832 4.9559418 4.551 .0000 .66666667E-01
T15 22.53676562 4.9405321 4.562 .0000 .66666667E-01
OUTPUT .8677267843 .15408184E-01 56.316 .0000 -1.1743092
FUEL -.4844835367 .36410849 -1.331 .1875 12.770359
LOAD -1.954404328 .44237771 -4.418 .0000 .56046015
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560929 , S.D.= 1.131971002 |
| Model size: Observations = 90, Parameters = 18, Deg.Fr.= 72 |
| Residuals: Sum of squares= 1.088190223 , Std.Dev.= .12294 |
| Fit: R-squared= .990458, Adjusted R-squared = .98820 |
| Model test: F[ 17, 72] = 439.62, Prob value = .00000 |
| Diagnostic: Log-L = 70.9837, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.010, Akaike Info. Crt.= -1.177 |
| Model does not contain ONE. R-squared and F can be negative! |
| Autocorrel: Durbin-Watson Statistic = 2.93900, Rho = -.46950 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
T1 20.49580478 4.2095283 4.869 .0000 .66666667E-01
T2 20.57803885 4.2215262 4.875 .0000 .66666667E-01
T3 20.65573100 4.2241771 4.890 .0000 .66666667E-01
T4 20.74075857 4.2457497 4.885 .0000 .66666667E-01
T5 21.19983202 4.4403312 4.774 .0000 .66666667E-01
T6 21.41162082 4.5386212 4.718 .0000 .66666667E-01
T7 21.50335085 4.5713968 4.704 .0000 .66666667E-01
T8 21.65402827 4.6228858 4.684 .0000 .66666667E-01
T9 21.82957108 4.6569062 4.688 .0000 .66666667E-01
T10 22.11380260 4.7926483 4.614 .0000 .66666667E-01
T11 22.46532734 4.9499089 4.539 .0000 .66666667E-01
T12 22.65133704 5.0085924 4.522 .0000 .66666667E-01
T13 22.61655508 4.9861391 4.536 .0000 .66666667E-01
T14 22.55222832 4.9559418 4.551 .0000 .66666667E-01
T15 22.53676562 4.9405321 4.562 .0000 .66666667E-01
OUTPUT .8677267843 .15408184E-01 56.316 .0000 -1.1743092
FUEL -.4844835367 .36410849 -1.331 .1875 12.770359
LOAD -1.954404328 .44237771 -4.418 .0000 .56046015
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
The following are the corresponding SAS REG procedure and STATA command for LSDV2 (outputs are skipped).
PROC REG DATA=masil.airline;
MODEL cost = t1-t15 output fuel load /NOINT;
RUN;
. regress cost t1-t15 output fuel load, noc
5.1.3 LSDV3 with a Restriction
In SAS, you need to use the RESTRICT statement to impose a restriction.
PROC REG DATA=masil.airline;
MODEL cost = t1-t15 output fuel load;
RESTRICT t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15=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 17 112.95270 6.64428 439.62 <.0001
Error 72 1.08819 0.01511
Corrected Total 89 114.04089
Root MSE 0.12294 R-Square 0.9905
Dependent Mean 13.36561 Adj R-Sq 0.9882
Coeff Var 0.91981
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 21.66698 4.62405 4.69 <.0001
t1 1 -1.17118 0.41783 -2.80 0.0065
t2 1 -1.08894 0.40586 -2.68 0.0090
t3 1 -1.01125 0.40323 -2.51 0.0144
t4 1 -0.92622 0.38177 -2.43 0.0178
t5 1 -0.46715 0.19076 -2.45 0.0168
t6 1 -0.25536 0.09856 -2.59 0.0116
t7 1 -0.16363 0.07190 -2.28 0.0258
t8 1 -0.01296 0.04862 -0.27 0.7907
t9 1 0.16259 0.06271 2.59 0.0115
t10 1 0.44682 0.17599 2.54 0.0133
t11 1 0.79834 0.32940 2.42 0.0179
t12 1 0.98435 0.38756 2.54 0.0132
t13 1 0.94957 0.36537 2.60 0.0113
t14 1 0.88524 0.33549 2.64 0.0102
t15 1 0.86978 0.32029 2.72 0.0083
output 1 0.86773 0.01541 56.32 <.0001
fuel 1 -0.48448 0.36411 -1.33 0.1875
load 1 -1.95440 0.44238 -4.42 <.0001
RESTRICT -1 -3.946E-15 . . .
* 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 17 112.95270 6.64428 439.62 <.0001
Error 72 1.08819 0.01511
Corrected Total 89 114.04089
Root MSE 0.12294 R-Square 0.9905
Dependent Mean 13.36561 Adj R-Sq 0.9882
Coeff Var 0.91981
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 21.66698 4.62405 4.69 <.0001
t1 1 -1.17118 0.41783 -2.80 0.0065
t2 1 -1.08894 0.40586 -2.68 0.0090
t3 1 -1.01125 0.40323 -2.51 0.0144
t4 1 -0.92622 0.38177 -2.43 0.0178
t5 1 -0.46715 0.19076 -2.45 0.0168
t6 1 -0.25536 0.09856 -2.59 0.0116
t7 1 -0.16363 0.07190 -2.28 0.0258
t8 1 -0.01296 0.04862 -0.27 0.7907
t9 1 0.16259 0.06271 2.59 0.0115
t10 1 0.44682 0.17599 2.54 0.0133
t11 1 0.79834 0.32940 2.42 0.0179
t12 1 0.98435 0.38756 2.54 0.0132
t13 1 0.94957 0.36537 2.60 0.0113
t14 1 0.88524 0.33549 2.64 0.0102
t15 1 0.86978 0.32029 2.72 0.0083
output 1 0.86773 0.01541 56.32 <.0001
fuel 1 -0.48448 0.36411 -1.33 0.1875
load 1 -1.95440 0.44238 -4.42 <.0001
RESTRICT -1 -3.946E-15 . . .
* Probability computed using beta distribution.
In STATA, define the restriction with the .constraint command and specify the restriction using the constraint() option of the .cnsreg command.
. constraint define 3 t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15=0
. cnsreg cost t1-t15 output fuel load, constraint(3)
Constrained
linear regression
Number of obs = 90
F( 17, 72) = 439.62
Prob > F = 0.0000
Root MSE = .12294
( 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]
-------------+----------------------------------------------------------------
t1 | -1.171179 .4178338 -2.80 0.007 -2.004115 -.3382422
t2 | -1.088945 .4058579 -2.68 0.009 -1.898008 -.2798816
t3 | -1.011252 .4032308 -2.51 0.014 -1.815078 -.2074266
t4 | -.9262249 .3817675 -2.43 0.018 -1.687265 -.1651852
t5 | -.4671515 .1907596 -2.45 0.017 -.8474239 -.0868791
t6 | -.2553627 .0985615 -2.59 0.012 -.4518415 -.0588839
t7 | -.1636326 .0718969 -2.28 0.026 -.3069564 -.0203088
t8 | -.0129552 .0486249 -0.27 0.791 -.1098872 .0839768
t9 | .1625876 .0627099 2.59 0.012 .0375776 .2875976
t10 | .4468191 .175994 2.54 0.013 .0959814 .7976568
t11 | .7983439 .3294027 2.42 0.018 .1416916 1.454996
t12 | .9843536 .3875583 2.54 0.013 .2117702 1.756937
t13 | .9495716 .3653675 2.60 0.011 .2212248 1.677918
t14 | .8852448 .3354912 2.64 0.010 .2164554 1.554034
t15 | .8697821 .3202933 2.72 0.008 .2312891 1.508275
output | .8677268 .0154082 56.32 0.000 .8370111 .8984424
fuel | -.4844835 .3641085 -1.33 0.188 -1.210321 .2413535
load | -1.954404 .4423777 -4.42 0.000 -2.836268 -1.07254
_cons | 21.66698 4.624053 4.69 0.000 12.4491 30.88486
------------------------------------------------------------------------------
F( 17, 72) = 439.62
Prob > F = 0.0000
Root MSE = .12294
( 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]
-------------+----------------------------------------------------------------
t1 | -1.171179 .4178338 -2.80 0.007 -2.004115 -.3382422
t2 | -1.088945 .4058579 -2.68 0.009 -1.898008 -.2798816
t3 | -1.011252 .4032308 -2.51 0.014 -1.815078 -.2074266
t4 | -.9262249 .3817675 -2.43 0.018 -1.687265 -.1651852
t5 | -.4671515 .1907596 -2.45 0.017 -.8474239 -.0868791
t6 | -.2553627 .0985615 -2.59 0.012 -.4518415 -.0588839
t7 | -.1636326 .0718969 -2.28 0.026 -.3069564 -.0203088
t8 | -.0129552 .0486249 -0.27 0.791 -.1098872 .0839768
t9 | .1625876 .0627099 2.59 0.012 .0375776 .2875976
t10 | .4468191 .175994 2.54 0.013 .0959814 .7976568
t11 | .7983439 .3294027 2.42 0.018 .1416916 1.454996
t12 | .9843536 .3875583 2.54 0.013 .2117702 1.756937
t13 | .9495716 .3653675 2.60 0.011 .2212248 1.677918
t14 | .8852448 .3354912 2.64 0.010 .2164554 1.554034
t15 | .8697821 .3202933 2.72 0.008 .2312891 1.508275
output | .8677268 .0154082 56.32 0.000 .8370111 .8984424
fuel | -.4844835 .3641085 -1.33 0.188 -1.210321 .2413535
load | -1.954404 .4423777 -4.42 0.000 -2.836268 -1.07254
_cons | 21.66698 4.624053 4.69 0.000 12.4491 30.88486
------------------------------------------------------------------------------
The following are the corresponding LIMDEP command for LSDV3 (outputs are skipped).
REGRESS;Lhs=COST;Rhs=ONE,T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,
T15,OUTPUT,FUEL,LOAD;
Cls:b(1)+b(2)+b(3)+b(4)+b(5)+b(6)+b(7)+b(8)+b(9)+b(10)+b(11)+b(12)
+b(13)+b(14)+b(15)=0$
The within effect mode for the fixed time effects needs to compute deviations from the time means. Keep in mind that the intercept should be suppressed.
5.2.1 Estimating the Time Effect Model
Let us manually estimate the fixed time effect model first.
. egen tm_cost = mean(cost), by(year) // compute time means
. egen tm_output = mean(output), by(year)
. egen tm_fuel = mean(fuel), by(year)
. egen tm_load = mean(load), by(year)
+---------------------------------------------------+
| year tm_cost tm_output tm_fuel tm_load |
|---------------------------------------------------|
| 1 12.36897 -1.790283 11.63606 .4788587 |
| 2 12.45963 -1.744389 11.66868 .4868322 |
| 3 12.60706 -1.577767 11.67494 .52358 |
| 4 12.77912 -1.443695 11.73193 .5244486 |
| 5 12.94143 -1.398122 12.26843 .5635266 |
| 6 13.0452 -1.393002 12.53826 .5541809 |
| 7 13.15965 -1.302416 12.62714 .5607425 |
| 8 13.29884 -1.222963 12.76768 .5670587 |
| 9 13.4651 -1.067003 12.86104 .6179098 |
| 10 13.70187 -.9023156 13.23183 .6233943 |
| 11 13.91324 -.9205539 13.66246 .5802577 |
| 12 14.05984 -.8641667 13.82315 .5856243 |
| 13 14.12841 -.7923916 13.75979 .5803183 |
| 14 14.23517 -.6428015 13.67403 .5804528 |
| 15 14.32062 -.5527684 13.62997 .5797168 |
+---------------------------------------------------+
| year tm_cost tm_output tm_fuel tm_load |
|---------------------------------------------------|
| 1 12.36897 -1.790283 11.63606 .4788587 |
| 2 12.45963 -1.744389 11.66868 .4868322 |
| 3 12.60706 -1.577767 11.67494 .52358 |
| 4 12.77912 -1.443695 11.73193 .5244486 |
| 5 12.94143 -1.398122 12.26843 .5635266 |
| 6 13.0452 -1.393002 12.53826 .5541809 |
| 7 13.15965 -1.302416 12.62714 .5607425 |
| 8 13.29884 -1.222963 12.76768 .5670587 |
| 9 13.4651 -1.067003 12.86104 .6179098 |
| 10 13.70187 -.9023156 13.23183 .6233943 |
| 11 13.91324 -.9205539 13.66246 .5802577 |
| 12 14.05984 -.8641667 13.82315 .5856243 |
| 13 14.12841 -.7923916 13.75979 .5803183 |
| 14 14.23517 -.6428015 13.67403 .5804528 |
| 15 14.32062 -.5527684 13.62997 .5797168 |
+---------------------------------------------------+
. gen tw_cost = cost - tm_cost // transform variables
. gen tw_output = output - tm_output
. gen tw_fuel = fuel - tm_fuel
. gen tw_load = load - tm_load
. regress tw_cost tw_output tw_fuel tw_load, noc // within time effect
Source |
SS df
MS
Number of obs = 90
-------------+------------------------------
F( 3, 87) = 2015.95
Model | 75.6459391 3
25.215313 Prob >
F = 0.0000
Residual | 1.08819023 87
.012507934
R-squared = 0.9858
-------------+------------------------------
Adj R-squared = 0.9853
Total | 76.7341294 90
.852601437 Root
MSE = .11184
------------------------------------------------------------------------------
tw_cost | Coef. Std.
Err. t
P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tw_output | .8677268 .0140171
61.90 0.000 .8398663
.8955873
tw_fuel | -.4844836 .3312359
-1.46 0.147 -1.142851 .1738836
tw_load | -1.954404 .4024388
-4.86 0.000 -2.754295 -1.154514
------------------------------------------------------------------------------
If you want to get intercepts of years, use 
. For example, the intercept of year 7 is 21.503=13.1597-{.8677*(-1.3024) + (-.4845)*12.6271 + (-1.9544)*.5607}. As discussed previously, the standard errors of the within effects model need to be adjusted. For instance, the correct standard error of fuel price is computed as .364 = .3312*sqrt(87/72).
. For example, the intercept of year 7 is 21.503=13.1597-{.8677*(-1.3024) + (-.4845)*12.6271 + (-1.9544)*.5607}. As discussed previously, the standard errors of the within effects model need to be adjusted. For instance, the correct standard error of fuel price is computed as .364 = .3312*sqrt(87/72).
5.2.2 Using the TSCSREG and PANEL procedures
You need to sort the data set by variables (i.e., year and airline) to appear in the ID statement of the TSCSREG and PANEL procedures.
PROC SORT DATA=masil.airline;
BY year airline;
PROC PANEL DATA=masil.airline;
ID year airline;
MODEL cost = output fuel load /FIXONE;
RUN;
The PANEL Procedure
Fixed One Way Estimates
Dependent Variable: cost
Model Description
Estimation Method FixOne
Number of Cross Sections 15
Time Series Length 6
Fit Statistics
SSE 1.0882 DFE 72
MSE 0.0151 Root MSE 0.1229
R-Square 0.9905
F Test for No Fixed Effects
Num DF Den DF F Value Pr > F
14 72 1.17 0.3178
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
CS1 1 -2.04096 0.7347 -2.78 0.0070 Cross Sectional
Effect 1
CS2 1 -1.95873 0.7228 -2.71 0.0084 Cross Sectional
Effect 2
CS3 1 -1.88103 0.7204 -2.61 0.0110 Cross Sectional
Effect 3
CS4 1 -1.79601 0.6988 -2.57 0.0122 Cross Sectional
Effect 4
CS5 1 -1.33693 0.5060 -2.64 0.0101 Cross Sectional
Effect 5
CS6 1 -1.12514 0.4086 -2.75 0.0075 Cross Sectional
Effect 6
CS7 1 -1.03341 0.3764 -2.75 0.0076 Cross Sectional
Effect 7
CS8 1 -0.88274 0.3260 -2.71 0.0085 Cross Sectional
Effect 8
CS9 1 -0.70719 0.2947 -2.40 0.0190 Cross Sectional
Effect 9
CS10 1 -0.42296 0.1668 -2.54 0.0134 Cross Sectional
Effect 10
CS11 1 -0.07144 0.0718 -1.00 0.3228 Cross Sectional
CS12 1 0.114571 0.0984 1.16 0.2482 Cross Sectional
Effect 12
CS13 1 0.079789 0.0844 0.95 0.3477 Cross Sectional
Effect 13
CS14 1 0.015463 0.0726 0.21 0.8320 Cross Sectional
Effect 14
Intercept 1 22.53677 4.9405 4.56 <.0001 Intercept
output 1 0.867727 0.0154 56.32 <.0001
fuel 1 -0.48448 0.3641 -1.33 0.1875
load 1 -1.9544 0.4424 -4.42 <.0001
Fixed One Way Estimates
Dependent Variable: cost
Model Description
Estimation Method FixOne
Number of Cross Sections 15
Time Series Length 6
Fit Statistics
SSE 1.0882 DFE 72
MSE 0.0151 Root MSE 0.1229
R-Square 0.9905
F Test for No Fixed Effects
Num DF Den DF F Value Pr > F
14 72 1.17 0.3178
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
CS1 1 -2.04096 0.7347 -2.78 0.0070 Cross Sectional
Effect 1
CS2 1 -1.95873 0.7228 -2.71 0.0084 Cross Sectional
Effect 2
CS3 1 -1.88103 0.7204 -2.61 0.0110 Cross Sectional
Effect 3
CS4 1 -1.79601 0.6988 -2.57 0.0122 Cross Sectional
Effect 4
CS5 1 -1.33693 0.5060 -2.64 0.0101 Cross Sectional
Effect 5
CS6 1 -1.12514 0.4086 -2.75 0.0075 Cross Sectional
Effect 6
CS7 1 -1.03341 0.3764 -2.75 0.0076 Cross Sectional
Effect 7
CS8 1 -0.88274 0.3260 -2.71 0.0085 Cross Sectional
Effect 8
CS9 1 -0.70719 0.2947 -2.40 0.0190 Cross Sectional
Effect 9
CS10 1 -0.42296 0.1668 -2.54 0.0134 Cross Sectional
Effect 10
CS11 1 -0.07144 0.0718 -1.00 0.3228 Cross Sectional
CS12 1 0.114571 0.0984 1.16 0.2482 Cross Sectional
Effect 12
CS13 1 0.079789 0.0844 0.95 0.3477 Cross Sectional
Effect 13
CS14 1 0.015463 0.0726 0.21 0.8320 Cross Sectional
Effect 14
Intercept 1 22.53677 4.9405 4.56 <.0001 Intercept
output 1 0.867727 0.0154 56.32 <.0001
fuel 1 -0.48448 0.3641 -1.33 0.1875
load 1 -1.9544 0.4424 -4.42 <.0001
The following TSCSREG procedure gives the same outputs.
PROC TSCSREG DATA=masil.airline;
ID year airline;
MODEL cost = output fuel load /FIXONE;
RUN;
The STATA .xtreg command uses the fe option for the fixed effect model.
. xtreg cost output fuel load, fe i(year)
Fixed-effects
(within)
regression
Number of obs
= 90
Group variable (i): year Number of groups = 15
R-sq: within = 0.9858 Obs per group: min = 6
between = 0.4812 avg = 6.0
overall = 0.5265 max = 6
F(3,72) = 1668.37
corr(u_i, Xb) = -0.1503 Prob > F = 0.0000
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .8677268 .0154082 56.32 0.000 .8370111 .8984424
fuel | -.4844835 .3641085 -1.33 0.188 -1.210321 .2413535
load | -1.954404 .4423777 -4.42 0.000 -2.836268 -1.07254
_cons | 21.66698 4.624053 4.69 0.000 12.4491 30.88486
-------------+----------------------------------------------------------------
sigma_u | .8027907
sigma_e | .12293801
rho | .97708602 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(14, 72) = 1.17 Prob > F = 0.3178
Group variable (i): year Number of groups = 15
R-sq: within = 0.9858 Obs per group: min = 6
between = 0.4812 avg = 6.0
overall = 0.5265 max = 6
F(3,72) = 1668.37
corr(u_i, Xb) = -0.1503 Prob > F = 0.0000
------------------------------------------------------------------------------
cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
output | .8677268 .0154082 56.32 0.000 .8370111 .8984424
fuel | -.4844835 .3641085 -1.33 0.188 -1.210321 .2413535
load | -1.954404 .4423777 -4.42 0.000 -2.836268 -1.07254
_cons | 21.66698 4.624053 4.69 0.000 12.4491 30.88486
-------------+----------------------------------------------------------------
sigma_u | .8027907
sigma_e | .12293801
rho | .97708602 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(14, 72) = 1.17 Prob > F = 0.3178
You need to pay attention to the Str=; subcommand for stratification.
--> REGRESS;Lhs=COST;Rhs=ONE,OUTPUT,FUEL,LOAD;Panel;Str=YEAR;Fixed$
+-----------------------------------------------------------------------+
| OLS Without Group Dummy Variables |
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560933 , S.D.= 1.131971444 |
| Model size: Observations = 90, Parameters = 4, Deg.Fr.= 86 |
| Residuals: Sum of squares= 1.335449522 , Std.Dev.= .12461 |
| Fit: R-squared= .988290, Adjusted R-squared = .98788 |
| Model test: F[ 3, 86] = 2419.33, Prob value = .00000 |
| Diagnostic: Log-L = 61.7699, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.122, Akaike Info. Crt.= -1.284 |
| Panel Data Analysis of COST [ONE way] |
| Unconditional ANOVA (No regressors) |
| Source Variation Deg. Free. Mean Square |
| Between 37.3068 14. 2.66477 |
| Residual 76.7341 75. 1.02312 |
| Total 114.041 89. 1.28136 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
OUTPUT .8827386341 .13254552E-01 66.599 .0000 -1.1743092
FUEL .4539777119 .20304240E-01 22.359 .0000 12.770359
LOAD -1.627507797 .34530293 -4.713 .0000 .56046016
Constant 9.516912231 .22924522 41.514 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+-----------------------------------------------------------------------+
| Least Squares with Group Dummy Variables |
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560933 , S.D.= 1.131971444 |
| Model size: Observations = 90, Parameters = 18, Deg.Fr.= 72 |
| Residuals: Sum of squares= 1.088193393 , Std.Dev.= .12294 |
| Fit: R-squared= .990458, Adjusted R-squared = .98820 |
| Model test: F[ 17, 72] = 439.62, Prob value = .00000 |
| Diagnostic: Log-L = 70.9836, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.010, Akaike Info. Crt.= -1.177 |
| Estd. Autocorrelation of e(i,t) .573531 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
OUTPUT .8677268093 .15408179E-01 56.316 .0000 -1.1743092
FUEL -.4844946699 .36410984 -1.331 .1868 12.770359
LOAD -1.954414378 .44237791 -4.418 .0000 .56046016
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+------------------------------------------------------------------------+
| Test Statistics for the Classical Model |
| |
| Model Log-Likelihood Sum of Squares R-squared |
| (1) Constant term only -138.35814 .1140409821D+03 .0000000 |
| (2) Group effects only -120.52864 .7673414157D+02 .3271354 |
| (3) X - variables only 61.76991 .1335449522D+01 .9882897 |
| (4) X and group effects 70.98362 .1088193393D+01 .9904579 |
| |
| Hypothesis Tests |
| Likelihood Ratio Test F Tests |
| Chi-squared d.f. Prob. F num. denom. Prob value |
| (2) vs (1) 35.659 14 .00117 2.605 14 75 .00404 |
| (3) vs (1) 400.256 3 .00000 2419.329 3 86 .00000 |
| (4) vs (1) 418.684 17 .00000 439.617 17 72 .00000 |
| (4) vs (2) 383.025 3 .00000 1668.364 3 72 .00000 |
| (4) vs (3) 18.427 14 .18800 1.169 14 72 .31776 |
+------------------------------------------------------------------------+
| OLS Without Group Dummy Variables |
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560933 , S.D.= 1.131971444 |
| Model size: Observations = 90, Parameters = 4, Deg.Fr.= 86 |
| Residuals: Sum of squares= 1.335449522 , Std.Dev.= .12461 |
| Fit: R-squared= .988290, Adjusted R-squared = .98788 |
| Model test: F[ 3, 86] = 2419.33, Prob value = .00000 |
| Diagnostic: Log-L = 61.7699, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.122, Akaike Info. Crt.= -1.284 |
| Panel Data Analysis of COST [ONE way] |
| Unconditional ANOVA (No regressors) |
| Source Variation Deg. Free. Mean Square |
| Between 37.3068 14. 2.66477 |
| Residual 76.7341 75. 1.02312 |
| Total 114.041 89. 1.28136 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
OUTPUT .8827386341 .13254552E-01 66.599 .0000 -1.1743092
FUEL .4539777119 .20304240E-01 22.359 .0000 12.770359
LOAD -1.627507797 .34530293 -4.713 .0000 .56046016
Constant 9.516912231 .22924522 41.514 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+-----------------------------------------------------------------------+
| Least Squares with Group Dummy Variables |
| Ordinary least squares regression Weighting variable = none |
| Dep. var. = COST Mean= 13.36560933 , S.D.= 1.131971444 |
| Model size: Observations = 90, Parameters = 18, Deg.Fr.= 72 |
| Residuals: Sum of squares= 1.088193393 , Std.Dev.= .12294 |
| Fit: R-squared= .990458, Adjusted R-squared = .98820 |
| Model test: F[ 17, 72] = 439.62, Prob value = .00000 |
| Diagnostic: Log-L = 70.9836, Restricted(b=0) Log-L = -138.3581 |
| LogAmemiyaPrCrt.= -4.010, Akaike Info. Crt.= -1.177 |
| Estd. Autocorrelation of e(i,t) .573531 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
OUTPUT .8677268093 .15408179E-01 56.316 .0000 -1.1743092
FUEL -.4844946699 .36410984 -1.331 .1868 12.770359
LOAD -1.954414378 .44237791 -4.418 .0000 .56046016
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+------------------------------------------------------------------------+
| Test Statistics for the Classical Model |
| |
| Model Log-Likelihood Sum of Squares R-squared |
| (1) Constant term only -138.35814 .1140409821D+03 .0000000 |
| (2) Group effects only -120.52864 .7673414157D+02 .3271354 |
| (3) X - variables only 61.76991 .1335449522D+01 .9882897 |
| (4) X and group effects 70.98362 .1088193393D+01 .9904579 |
| |
| Hypothesis Tests |
| Likelihood Ratio Test F Tests |
| Chi-squared d.f. Prob. F num. denom. Prob value |
| (2) vs (1) 35.659 14 .00117 2.605 14 75 .00404 |
| (3) vs (1) 400.256 3 .00000 2419.329 3 86 .00000 |
| (4) vs (1) 418.684 17 .00000 439.617 17 72 .00000 |
| (4) vs (2) 383.025 3 .00000 1668.364 3 72 .00000 |
| (4) vs (3) 18.427 14 .18800 1.169 14 72 .31776 |
+------------------------------------------------------------------------+
The between effect model regresses time means of dependent variables on those of independent variables. See also 3.2 and 4.6.
. collapse (mean) tm_cost=cost (mean) tm_output=output (mean) tm_fuel=fuel (mean) tm_load=load, by(year)
. regress tm_cost tm_output tm_fuel tm_load // between time effect
Source |
SS df
MS
Number of obs = 15
-------------+------------------------------ F( 3, 11) = 4074.33
Model | 6.21220479 3 2.07073493 Prob > F = 0.0000
Residual | .005590631 11 .000508239 R-squared = 0.9991
-------------+------------------------------ Adj R-squared = 0.9989
Total | 6.21779542 14 .444128244 Root MSE = .02254
------------------------------------------------------------------------------
tm_cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tm_output | 1.133337 .0512898 22.10 0.000 1.020449 1.246225
tm_fuel | .3342486 .0228284 14.64 0.000 .2840035 .3844937
tm_load | -1.350727 .2478264 -5.45 0.000 -1.896189 -.8052644
_cons | 11.18505 .3660016 30.56 0.000 10.37949 11.99062
------------------------------------------------------------------------------
-------------+------------------------------ F( 3, 11) = 4074.33
Model | 6.21220479 3 2.07073493 Prob > F = 0.0000
Residual | .005590631 11 .000508239 R-squared = 0.9991
-------------+------------------------------ Adj R-squared = 0.9989
Total | 6.21779542 14 .444128244 Root MSE = .02254
------------------------------------------------------------------------------
tm_cost | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
tm_output | 1.133337 .0512898 22.10 0.000 1.020449 1.246225
tm_fuel | .3342486 .0228284 14.64 0.000 .2840035 .3844937
tm_load | -1.350727 .2478264 -5.45 0.000 -1.896189 -.8052644
_cons | 11.18505 .3660016 30.56 0.000 10.37949 11.99062
------------------------------------------------------------------------------
The SAS PANEL procedure has the /BTWNT option to estimate the between effect model.
PROC PANEL DATA=masil.airline;
ID airline year;
MODEL cost = output fuel load /BTWNT;
RUN;
The PANEL Procedure
Between Time Periods Estimates
Dependent Variable: cost
Model Description
Estimation Method BtwTime
Number of Cross Sections 6
Time Series Length 15
Fit Statistics
SSE 0.0056 DFE 11
MSE 0.0005 Root MSE 0.0225
R-Square 0.9991
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 11.18504 0.3660 30.56 <.0001 Intercept
output 1 1.133335 0.0513 22.10 <.0001
fuel 1 0.334249 0.0228 14.64 <.0001
load 1 -1.35073 0.2478 -5.45 0.0002
Between Time Periods Estimates
Dependent Variable: cost
Model Description
Estimation Method BtwTime
Number of Cross Sections 6
Time Series Length 15
Fit Statistics
SSE 0.0056 DFE 11
MSE 0.0005 Root MSE 0.0225
R-Square 0.9991
Parameter Estimates
Standard
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 11.18504 0.3660 30.56 <.0001 Intercept
output 1 1.133335 0.0513 22.10 <.0001
fuel 1 0.334249 0.0228 14.64 <.0001
load 1 -1.35073 0.2478 -5.45 0.0002
You may use the be option in the STATA .xtreg command and the Means; subcommand in LIMDEP (outputs are skipped).
. xtreg cost output fuel load, be i(year) // between time effect model
--> REGRESS;Lhs=COST;Rhs=ONE,OUTPUT,FUEL,LOAD;Panel;Str=YEAR;Means$
5.4 Testing Fixed Time Effects.
The null hypothesis is that all time dummy parameters except one are zero. The F statistic is 
. The p-value of .3180 does not reject the null hypothesis.
. The p-value of .3180 does not reject the null hypothesis.
The SAS TSCSREG and PANEL procedures and the STATA .xtreg command conduct the Wald test. You may get the same test using the TEST statement in LSDV1 and the STATA .test command (the output is skipped).
PROC REG DATA=masil.airline;
MODEL cost = t1-t14 output fuel load;
TEST t1=t2=t3=t4=t5=t6=t7=t8=t9=t10=t11=t12=t13=t14=0;
RUN;
. quietly regress cost t1-t14 output fuel load
. test t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14
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