Table 3.3, page 54.
use https://stats.idre.ucla.edu/stat/stata/examples/chp/p054, clear
list y-x3
y x1 x2 x3
1. 43 51 30 39
2. 63 64 51 54
3. 71 70 68 69
4. 61 63 45 47
..
[remainder of output omitted]
list x4-x6
x4 x5 x6
1. 61 92 45
2. 63 73 47
3. 76 86 48
4. 54 84 35
..
[remainder of output omitted]
Coefficients for equation 3.12, page 57.
regress y x1 x2
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 2, 27) = 29.10
Model | 2935.10281 2 1467.55141 Prob > F = 0.0000
Residual | 1361.86385 27 50.4394019 R-squared = 0.6831
---------+------------------------------ Adj R-squared = 0.6596
Total | 4296.96667 29 148.171264 Root MSE = 7.1021
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .7803434 .1193876 6.536 0.000 .5353802 1.025307
x2 | -.0501598 .1299192 -0.386 0.702 -.316732 .2164124
_cons | 15.32762 7.160234 2.141 0.041 .6360331 30.01921
------------------------------------------------------------------------------
Coefficients for equation 3.13, page 57.
regress y x1
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 1, 28) = 59.86
Model | 2927.58425 1 2927.58425 Prob > F = 0.0000
Residual | 1369.38241 28 48.9065148 R-squared = 0.6813
---------+------------------------------ Adj R-squared = 0.6699
Total | 4296.96667 29 148.171264 Root MSE = 6.9933
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .7546098 .0975329 7.737 0.000 .5548227 .9543969
_cons | 14.37632 6.619986 2.172 0.039 .8158929 27.93675
------------------------------------------------------------------------------
Compute residuals for this model.
Note: The variable eyx1 is set to type double for greater precision.
predict double eyx1, residual
Coefficients for equation 3.14, page 57.
regress x2 x1
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 1, 28) = 12.68
Model | 1353.17314 1 1353.17314 Prob > F = 0.0013
Residual | 2988.29352 28 106.724769 R-squared = 0.3117
---------+------------------------------ Adj R-squared = 0.2871
Total | 4341.46667 29 149.705747 Root MSE = 10.331
------------------------------------------------------------------------------
x2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .513032 .1440789 3.561 0.001 .2178997 .8081642
_cons | 18.9654 9.779267 1.939 0.063 -1.066517 38.99732
------------------------------------------------------------------------------
Compute residuals for this model.
predict double ex2x1, residual
Table 3.4, page 58.
list eyx1 ex2x1
eyx1 ex2x1
1. -9.861421 -15.13003
2. .3286522 -.7994502
3. 3.800993 13.12236
4. -.916738 -6.286418
5. 7.764115 -2.981898
6. -12.87986 1.817838
7. -6.935177 -11.33855
8. .0279442 -7.442802
9. -4.254324 10.96597
10. 6.592482 -5.260354
11. 9.62936 6.843902
12. 7.347092 -2.747322
13. 7.837872 6.226614
14. -9.008934 21.45294
15. 4.518724 -4.468866
16. -1.291203 -15.13828
17. -4.518154 1.426878
18. 5.347092 15.25268
19. -2.199007 -8.877642
20. -8.143689 19.27874
21. 5.439288 -6.486683
22. 3.592482 1.739646
23. -11.18057 -.8255141
24. -2.296883 4.052413
25. 7.874751 -4.66913
26. -6.481276 7.531134
27. 7.027944 .557198
28. -9.389079 -4.208226
29. 6.481846 8.426878
30. 5.745676 -22.03403
Coefficients for equation 3.15, page 58.
Note: Values such as -9.19e-15 are zero as represented by Stata in double precision.
regress eyx1 ex2x1
Source | SS df MS Number of obs = 30
-------------+------------------------------ F( 1, 28) = 0.15
Model | 7.51856161 1 7.51856161 Prob > F = 0.6972
Residual | 1361.86385 28 48.6379947 R-squared = 0.0055
-------------+------------------------------ Adj R-squared = -0.0300
Total | 1369.38241 29 47.2200832 Root MSE = 6.9741
------------------------------------------------------------------------------
eyx1 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ex2x1 | -.0501598 .1275781 -0.39 0.697 -.3114917 .2111721
_cons | -9.19e-15 1.27329 -0.00 1.000 -2.608216 2.608216
------------------------------------------------------------------------------
Equation 3.25, page 62 and table 3.5, page 63.
regress y x1 x2 x3 x4 x5 x6
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 6, 23) = 10.50
Model | 3147.96634 6 524.661057 Prob > F = 0.0000
Residual | 1149.00032 23 49.9565359 R-squared = 0.7326
---------+------------------------------ Adj R-squared = 0.6628
Total | 4296.96667 29 148.171264 Root MSE = 7.068
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6131876 .1609831 3.809 0.001 .2801687 .9462065
x2 | -.0730501 .1357247 -0.538 0.596 -.353818 .2077178
x3 | .3203321 .1685203 1.901 0.070 -.0282787 .6689429
x4 | .0817321 .2214777 0.369 0.715 -.3764293 .5398936
x5 | .0383814 .1469954 0.261 0.796 -.2657018 .3424647
x6 | -.2170567 .1782095 -1.218 0.236 -.585711 .1515977
_cons | 10.78708 11.58926 0.931 0.362 -13.18713 34.76128
------------------------------------------------------------------------------
Table 3.7, page 67.
test x1 x2 x3 x4 x5 x6
( 1) x1 = 0.0
( 2) x2 = 0.0
( 3) x3 = 0.0
( 4) x4 = 0.0
( 5) x5 = 0.0
( 6) x6 = 0.0
F( 6, 23) = 10.50
Prob > F = 0.0000
Equation 3.40, page 68.
test x2 x4 x5 x6
( 1) x2 = 0.0
( 2) x4 = 0.0
( 3) x5 = 0.0
( 4) x6 = 0.0
F( 4, 23) = 0.53
Prob > F = 0.7158
Table 3.8, page 69.
regress y x1 x3
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 2, 27) = 32.74
Model | 3042.3177 2 1521.15885 Prob > F = 0.0000
Residual | 1254.64897 27 46.4684804 R-squared = 0.7080
---------+------------------------------ Adj R-squared = 0.6864
Total | 4296.96667 29 148.171264 Root MSE = 6.8168
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6435176 .1184774 5.432 0.000 .400422 .8866132
x3 | .2111918 .1344037 1.571 0.128 -.0645818 .4869655
_cons | 9.87088 7.061224 1.398 0.174 -4.617553 24.35931
------------------------------------------------------------------------------
Equation following 3.48, page 71.
generate w = x1 + x3
regress y w
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 1, 28) = 56.46
Model | 2872.37206 1 2872.37206 Prob > F = 0.0000
Residual | 1424.59461 28 50.8783788 R-squared = 0.6685
---------+------------------------------ Adj R-squared = 0.6566
Total | 4296.96667 29 148.171264 Root MSE = 7.1329
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
w | .4443897 .0591439 7.514 0.000 .3232388 .5655406
_cons | 9.988214 7.38841 1.352 0.187 -5.146257 25.12268
------------------------------------------------------------------------------
F-ratio middle of page 71.
regress y x1 x3
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 2, 27) = 32.74
Model | 3042.3177 2 1521.15885 Prob > F = 0.0000
Residual | 1254.64897 27 46.4684804 R-squared = 0.7080
---------+------------------------------ Adj R-squared = 0.6864
Total | 4296.96667 29 148.171264 Root MSE = 6.8168
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6435176 .1184774 5.432 0.000 .400422 .8866132
x3 | .2111918 .1344037 1.571 0.128 -.0645818 .4869655
_cons | 9.87088 7.061224 1.398 0.174 -4.617553 24.35931
------------------------------------------------------------------------------
test x1 = x3
( 1) x1 - x3 = 0.0
F( 1, 27) = 3.66
Prob > F = 0.0665
F-ratio near the top of page 72.
Note: The accum option allows you to accumulate the simultaneous results of of consecutive test commands.
regress y x1 x2 x3 x4 x5 x6
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 6, 23) = 10.50
Model | 3147.96634 6 524.661057 Prob > F = 0.0000
Residual | 1149.00032 23 49.9565359 R-squared = 0.7326
---------+------------------------------ Adj R-squared = 0.6628
Total | 4296.96667 29 148.171264 Root MSE = 7.068
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6131876 .1609831 3.809 0.001 .2801687 .9462065
x2 | -.0730501 .1357247 -0.538 0.596 -.353818 .2077178
x3 | .3203321 .1685203 1.901 0.070 -.0282787 .6689429
x4 | .0817321 .2214777 0.369 0.715 -.3764293 .5398936
x5 | .0383814 .1469954 0.261 0.796 -.2657018 .3424647
x6 | -.2170567 .1782095 -1.218 0.236 -.585711 .1515977
_cons | 10.78708 11.58926 0.931 0.362 -13.18713 34.76128
------------------------------------------------------------------------------
test x1=x3
( 1) x1 - x3 = 0.0
F( 1, 23) = 1.21
Prob > F = 0.2821
test x2 x4 x5 x6, accum
( 1) x1 - x3 = 0.0
( 2) x2 = 0.0
( 3) x4 = 0.0
( 4) x5 = 0.0
( 5) x6 = 0.0
F( 5, 23) = 1.10
Prob > F = 0.3857
Method 1: Equations at the bottom of page 72 and top of page 73.
generate yprime = y - x3
generate v = x1 - x3
regress yprime v
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 1, 28) = 37.79
Model | 1794.31921 1 1794.31921 Prob > F = 0.0000
Residual | 1329.54746 28 47.4838377 R-squared = 0.5744
---------+------------------------------ Adj R-squared = 0.5592
Total | 3123.86667 29 107.71954 Root MSE = 6.8909
------------------------------------------------------------------------------
yprime | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
v | .6938232 .1128683 6.147 0.000 .462623 .9250233
_cons | 1.166543 1.707882 0.683 0.500 -2.331896 4.664982
------------------------------------------------------------------------------
display 1 - .6938232
.3061768
Method 2: Equations at the bottom of page 72 and top of page 73. cnsreg, constrained linear regression, computes the coefficients for both x1 and x3 in one command.
constraint define 1 x1 + x3 = 1
cnsreg y x1 x3, c(1)
Constrained linear regression Number of obs = 30
Root MSE = 6.8909
( 1) x1 + x3 = 1.0
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6938232 .1128683 6.147 0.000 .462623 .9250233
x3 | .3061768 .1128683 2.713 0.011 .0749767 .537377
_cons | 1.166543 1.707882 0.683 0.500 -2.331896 4.664982
------------------------------------------------------------------------------
F-ratio at the top of page 73.
regress y x1 x3
Source | SS df MS Number of obs = 30
---------+------------------------------ F( 2, 27) = 32.74
Model | 3042.3177 2 1521.15885 Prob > F = 0.0000
Residual | 1254.64897 27 46.4684804 R-squared = 0.7080
---------+------------------------------ Adj R-squared = 0.6864
Total | 4296.96667 29 148.171264 Root MSE = 6.8168
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x1 | .6435176 .1184774 5.432 0.000 .400422 .8866132
x3 | .2111918 .1344037 1.571 0.128 -.0645818 .4869655
_cons | 9.87088 7.061224 1.398 0.174 -4.617553 24.35931
------------------------------------------------------------------------------
test x1 + x3 = 1
( 1) x1 + x3 = 1.0
F( 1, 27) = 1.61
Prob > F = 0.2151