For each of the following problems, you are given some independent X variables and a dependent Y variable. Paste the data into Excel or another statistics program, and perform a multiple regression calculation with the given variables. The answers follow the data.
. X1 X2 X3 X4 Y1
1: 88 1 25 1 720
2: 74 57 48 3 1522
3: 18 21 80 6 -5
4: 65 53 76 11 1152
5: 6 87 31 11 1602
6: 68 84 100 4 1589
7: 66 32 97 6 559
8: 1 15 99 13 -426
9: 69 61 28 6 1721
10: 2 93 25 2 1703
11: 54 33 1 3 1218
12: 71 3 28 5 578
13: 31 59 24 0 1315
14: 22 89 98 7 1254
15: 6 47 54 4 597
r squared: 1.0000
Standard error: ....
Coefficient 1 X1 10.0000
t statistic: ...
Standard err: 0.0000
Coefficient 2 X2 20.0000
t statistic: ...
Standard err: 0.0000
Coefficient 3 X3 -8.0000
t statistic: ...
Standard err: 0.0000
Coefficient 4 X4 3.0000
t statistic: ...
Standard err: 0.0000
Coefficient 5 Constant 17.0000
t statistic: ...
Standard err: 0.0000
. X1 X2 X3 Y1
1: 88 1 25 720
2: 74 57 48 1522
3: 18 21 80 -5
4: 65 53 76 1152
5: 6 87 31 1602
6: 68 84 100 1589
7: 66 32 97 559
8: 1 15 99 -426
9: 69 61 28 1721
10: 2 93 25 1703
11: 54 33 1 1218
12: 71 3 28 578
13: 31 59 24 1315
14: 22 89 98 1254
15: 6 47 54 597
r squared: 0.9998
Standard error: 1.0497969445E+01
Coefficient 1 X1 9.8952
t statistic: 103.1190
Standard err: 0.0960
Coefficient 2 X2 19.9569
t statistic: 207.9134
Standard err: 0.0960
Coefficient 3 X3 -7.8432
t statistic: -94.8108
Standard err: 0.0827
Coefficient 4 Constant 31.4800
t statistic: 3.4851
Standard err: 9.0328
F statistic:17800.4965 3, 11 degrees of freedom
Same as problem 1, except variable X4 has been left out. There still is
a very good fit.
. X1 X3 X4 Y1
1: 88 25 1 720
2: 74 48 3 1522
3: 18 80 6 -5
4: 65 76 11 1152
5: 6 31 11 1602
6: 68 100 4 1589
7: 66 97 6 559
8: 1 99 13 -426
9: 69 28 6 1721
10: 2 25 2 1703
11: 54 1 3 1218
12: 71 28 5 578
13: 31 24 0 1315
14: 22 98 7 1254
15: 6 54 4 597
r squared: 0.2018
Standard error: 6.5355988994E+02
Coefficient 1 X1 2.7627
t statistic: 0.4683
Standard err: 5.8988
Coefficient 2 X3 -6.1229
t statistic: -1.0369
Standard err: 5.9048
Coefficient 3 X4 -22.0526
t statistic: -0.3952
Standard err: 55.8036
Coefficient 4 Constant 1341.3670
t statistic: 2.7929
Standard err: 480.2700
F statistic: 0.9270 3, 11 degrees of freedom
Same as problem 1, except variable X2 is left out. The fit is much worse.
. X2 X3 X4 Y1
1: 1 25 1 720
2: 57 48 3 1522
3: 21 80 6 -5
4: 53 76 11 1152
5: 87 31 11 1602
6: 84 100 4 1589
7: 32 97 6 559
8: 15 99 13 -426
9: 61 28 6 1721
10: 93 25 2 1703
11: 33 1 3 1218
12: 3 28 5 578
13: 59 24 0 1315
14: 89 98 7 1254
15: 47 54 4 597
r squared: 0.8166
Standard error: 3.1330833087E+02
Coefficient 1 X2 16.6736
t statistic: 6.1498
Standard err: 2.7113
Coefficient 2 X3 -7.3966
t statistic: -2.6064
Standard err: 2.8379
Coefficient 3 X4 -25.0014
t statistic: -0.9750
Standard err: 25.6413
Coefficient 4 Constant 727.6567
t statistic: 3.4793
Standard err: 209.1405
F statistic: 16.3222 3, 11 degrees of freedom
Same as problem 1, except without X1.
. X1 X2 Y1
1: 88 1 720
2: 74 57 1522
3: 18 21 -5
4: 65 53 1152
5: 6 87 1602
6: 68 84 1589
7: 66 32 559
8: 1 15 -426
9: 69 61 1721
10: 2 93 1703
11: 54 33 1218
12: 71 3 578
13: 31 59 1315
14: 22 89 1254
15: 6 47 597
r squared: 0.8315
Standard error: 2.8750017921E+02
Coefficient 1 X1 10.9014
t statistic: 4.1739
Standard err: 2.6118
Coefficient 2 X2 19.6518
t statistic: 7.4800
Standard err: 2.6272
Coefficient 3 Constant -422.1920
t statistic: -2.0122
Standard err: 209.8144
F statistic: 29.6080 2, 12 degrees of freedom
. X1 X4 Y1
1: 88 1 720
2: 74 3 1522
3: 18 6 -5
4: 65 11 1152
5: 6 11 1602
6: 68 4 1589
7: 66 6 559
8: 1 13 -426
9: 69 6 1721
10: 2 2 1703
11: 54 3 1218
12: 71 5 578
13: 31 0 1315
14: 22 7 1254
15: 6 4 597
r squared: 0.1238
Standard error: 6.5560542681E+02
Coefficient 1 X1 2.5604
t statistic: 0.4329
Standard err: 5.9141
Coefficient 2 X4 -50.4097
t statistic: -1.0331
Standard err: 48.7957
Coefficient 3 Constant 1172.7597
t statistic: 2.5870
Standard err: 453.3214
F statistic: 0.8476 2, 12 degrees of freedom
Problems 1 to 6 illustrate what can happen when independent variables are left
out of a regression calculation. The true relationship between X1, X2, X3, X4,
and Y1 is given in problem 1. In problems 2 to 6, at least one independent variable
is left out. In some cases the omitted variable does not seriously affect the
results, but in other cases the omitted variable causes the coefficient
estimates to be far off their true value.
X1 Y
1: 5 -152
2: 39 1768
3: 36 -342
4: 47 878
5: 53 248
6: 99 -272
7: 67 1068
8: 90 148
9: 100 -382
10: 48 -302
11: 74 528
12: 43 758
13: 32 -132
14: 86 728
15: 36 538
16: 8 1358
17: 47 118
18: 96 368
19: 67 1578
20: 78 -52
r squared: 0.0366
Standard error: 6.6160524348E+02
Coefficient 1 X1 -4.4062
t statistic: -0.8269
Standard err: 5.3284
Coefficient 2 Constant 676.0743
t statistic: 1.9857
Standard err: 340.4720
F statistic: 0.6838 1, 18 degrees of freedom
. X2 Y
1: 7 -152
2: 39 1768
3: 5 -342
4: 32 878
5: 17 248
6: 8 -272
7: 22 1068
8: 11 148
9: 4 -382
10: 7 -302
11: 18 528
12: 25 758
13: 12 -132
14: 15 728
15: 22 538
16: 34 1358
17: 5 118
18: 22 368
19: 33 1578
20: 3 -52
r squared: 0.8652
Standard error: 2.4745819242E+02
Coefficient 1 X2 54.3787
t statistic: 10.7496
Standard err: 5.0587
Coefficient 2 Constant -504.6568
t statistic: -4.9247
Standard err: 102.4737
F statistic: 115.5547 1, 18 degrees of freedom
. X3 Y
1: 52 -152
2: 20 1768
3: 61 -342
4: 74 878
5: 62 248
6: 69 -272
7: 5 1068
8: 42 148
9: 60 -382
10: 67 -302
11: 39 528
12: 51 758
13: 75 -132
14: 4 728
15: 58 538
16: 36 1358
17: 15 118
18: 75 368
19: 9 1578
20: 22 -52
r squared: 0.2977
Standard error: 5.6486544952E+02
D-W: 2.5574
Coefficient 1 X3 -14.5631
t statistic: -2.7625
Standard err: 5.2717
Coefficient 2 Constant 1074.9274
t statistic: 4.0135
Standard err: 267.8266
F statistic: 7.6314 1, 18 degrees of freedom
. X4 Y
1: 8 -152
2: 42 1768
3: 44 -342
4: 37 878
5: 19 248
6: 13 -272
7: 33 1068
8: 8 148
9: 25 -382
10: 9 -302
11: 8 528
12: 30 758
13: 34 -132
14: 18 728
15: 2 538
16: 5 1358
17: 25 118
18: 35 368
19: 40 1578
20: 36 -52
r squared: 0.0467
Standard error: 6.5811943916E+02
Coefficient 1 X4 10.2332
t statistic: 0.9393
Standard err: 10.8948
Coefficient 2 Constant 181.5077
t statistic: 0.6137
Standard err: 295.7803
F statistic: 0.8822 1, 18 degrees of freedom
. X1 X2 Y
1: 5 7 -152
2: 39 39 1768
3: 36 5 -342
4: 47 32 878
5: 53 17 248
6: 99 8 -272
7: 67 22 1068
8: 90 11 148
9: 100 4 -382
10: 48 7 -302
11: 74 18 528
12: 43 25 758
13: 32 12 -132
14: 86 15 728
15: 36 22 538
16: 8 34 1358
17: 47 5 118
18: 96 22 368
19: 67 33 1578
20: 78 3 -52
r squared: 0.8676
Standard error: 2.5241212684E+02
Coefficient 1 X1 1.1526
t statistic: 0.5481
Standard err: 2.1029
Coefficient 2 X2 55.1274
t statistic: 10.3279
Standard err: 5.3377
Coefficient 3 Constant -583.7523
t statistic: -3.2760
Standard err: 178.1923
F statistic: 55.6819 2, 17 degrees of freedom
. X1 X3 Y
1: 5 52 -152
2: 39 20 1768
3: 36 61 -342
4: 47 74 878
5: 53 62 248
6: 99 69 -272
7: 67 5 1068
8: 90 42 148
9: 100 60 -382
10: 48 67 -302
11: 74 39 528
12: 43 51 758
13: 32 75 -132
14: 86 4 728
15: 36 58 538
16: 8 36 1358
17: 47 15 118
18: 96 75 368
19: 67 9 1578
20: 78 22 -52
r squared: 0.3516
Standard error: 5.5849940111E+02
D-W: 2.7405
Coefficient 1 X1 -5.3608
t statistic: -1.1886
Standard err: 4.5103
Coefficient 2 X3 -15.0206
t statistic: -2.8739
Standard err: 5.2265
Coefficient 3 Constant 1403.9349
t statistic: 3.6649
Standard err: 383.0766
F statistic: 4.6095 2, 17 degrees of freedom
. X1 X4 Y
1: 5 8 -152
2: 39 42 1768
3: 36 44 -342
4: 47 37 878
5: 53 19 248
6: 99 13 -272
7: 67 33 1068
8: 90 8 148
9: 100 25 -382
10: 48 9 -302
11: 74 8 528
12: 43 30 758
13: 32 34 -132
14: 86 18 728
15: 36 2 538
16: 8 5 1358
17: 47 25 118
18: 96 35 368
19: 67 40 1578
20: 78 36 -52
*:MR X1,X4,Y
r squared: 0.0919
Standard error: 6.6095574632E+02
Coefficient 1 X1 -4.9175
t statistic: -0.9197
Standard err: 5.3469
Coefficient 2 X4 11.1832
t statistic: 1.0175
Standard err: 10.9904
Coefficient 3 Constant 442.1377
t statistic: 1.0769
Standard err: 410.5476
F statistic: 0.8603 2, 17 degrees of freedom
. X2 X3 Y
1: 7 52 -152
2: 39 20 1768
3: 5 61 -342
4: 32 74 878
5: 17 62 248
6: 8 69 -272
7: 22 5 1068
8: 11 42 148
9: 4 60 -382
10: 7 67 -302
11: 18 39 528
12: 25 51 758
13: 12 75 -132
14: 15 4 728
15: 22 58 538
16: 34 36 1358
17: 5 15 118
18: 22 75 368
19: 33 9 1578
20: 3 22 -52
r squared: 1.0000
Standard error: 8.4772325116E-09
Coefficient 1 X2 50.0000
t statistic: 282700885650.0000
Standard err: 0.0000
Coefficient 2 X3 -10.0000
t statistic: -123846573050.0000
Standard err: 0.0000
Coefficient 3 Constant 18.0000
t statistic: 3279052617.3000
Standard err: 0.0000
F statistic: 0.0000 2, 17 degrees of freedom
. X2 X4 Y
1: 7 8 -152
2: 39 42 1768
3: 5 44 -342
4: 32 37 878
5: 17 19 248
6: 8 13 -272
7: 22 33 1068
8: 11 8 148
9: 4 25 -382
10: 7 9 -302
11: 18 8 528
12: 25 30 758
13: 12 34 -132
14: 15 18 728
15: 22 2 538
16: 34 5 1358
17: 5 25 118
18: 22 35 368
19: 33 40 1578
20: 3 36 -52
*:MR X2,X4,Y
r squared: 0.8664
Standard error: 2.5353139435E+02
Coefficient 1 X2 53.9793
t statistic: 10.2122
Standard err: 5.2858
Coefficient 2 X4 1.6466
t statistic: 0.3847
Standard err: 4.2805
Coefficient 3 Constant -536.6235
t statistic: -4.0077
Standard err: 133.8977
F statistic: 55.1164 2, 17 degrees of freedom
. X3 X4 Y
1: 52 8 -152
2: 20 42 1768
3: 61 44 -342
4: 74 37 878
5: 62 19 248
6: 69 13 -272
7: 5 33 1068
8: 42 8 148
9: 60 25 -382
10: 67 9 -302
11: 39 8 528
12: 51 30 758
13: 75 34 -132
14: 4 18 728
15: 58 2 538
16: 36 5 1358
17: 15 25 118
18: 75 35 368
19: 9 40 1578
20: 22 36 -52
r squared: 0.3194
Standard error: 5.7219800395E+02
D-W: 2.5558
Coefficient 1 X3 -14.0538
t statistic: -2.6099
Standard err: 5.3848
Coefficient 2 X4 7.0296
t statistic: 0.7360
Standard err: 9.5517
Coefficient 3 Constant 886.5653
t statistic: 2.3770
Standard err: 372.9779
F statistic: 3.9894 2, 17 degrees of freedom
. X1 X2 X3 Y
1: 5 7 52 -152
2: 39 39 20 1768
3: 36 5 61 -342
4: 47 32 74 878
5: 53 17 62 248
6: 99 8 69 -272
7: 67 22 5 1068
8: 90 11 42 148
9: 100 4 60 -382
10: 48 7 67 -302
11: 74 18 39 528
12: 43 25 51 758
13: 32 12 75 -132
14: 86 15 4 728
15: 36 22 58 538
16: 8 34 36 1358
17: 47 5 15 118
18: 96 22 75 368
19: 67 33 9 1578
20: 78 3 22 -52
r squared: 1.0000
Standard error: 1.3170439975E-08
Coefficient 1 X1 -0.0000
t statistic: -0.6327
Standard err: 0.0000
Coefficient 2 X2 50.0000
t statistic: 174842452260.0000
Standard err: 0.0000
Coefficient 3 X3 -10.0000
t statistic: -79019521150.0000
Standard err: 0.0000
Coefficient 4 Constant 18.0000
t statistic: 1497711452.4000
Standard err: 0.0000
F statistic: 0.0000 3, 16 degrees of freedom
. X1 X2 X4 Y 1: 5 7 8 -152 2: 39 39 42 1768 3: 36 5 44 -342 4: 47 32 37 878 5: 53 17 19 248 6: 99 8 13 -272 7: 67 22 33 1068 8: 90 11 8 148 9: 100 4 25 -382 10: 48 7 9 -302 11: 74 18 8 528 12: 43 25 30 758 13: 32 12 34 -132 14: 86 15 18 728 15: 36 22 2 538 16: 8 34 5 1358 17: 47 5 25 118 18: 96 22 35 368 19: 67 33 40 1578 20: 78 3 36 -52 ****************************************
. X1 X3 X4 Y
1: 5 52 8 -152
2: 39 20 42 1768
3: 36 61 44 -342
4: 47 74 37 878
5: 53 62 19 248
6: 99 69 13 -272
7: 67 5 33 1068
8: 90 42 8 148
9: 100 60 25 -382
10: 48 67 9 -302
11: 74 39 8 528
12: 43 51 30 758
13: 32 75 34 -132
14: 86 4 18 728
15: 36 58 2 538
16: 8 36 5 1358
17: 47 15 25 118
18: 96 75 35 368
19: 67 9 40 1578
20: 78 22 36 -52
r squared: 0.3797
Standard error: 5.6306381919E+02
Coefficient 1 X1 -5.6930
t statistic: -1.2474
Standard err: 4.5638
Coefficient 2 X3 -14.4668
t statistic: -2.7249
Standard err: 5.3092
Coefficient 3 X4 8.0353
t statistic: 0.8518
Standard err: 9.4337
Coefficient 4 Constant 1209.0138
t statistic: 2.6932
Standard err: 448.9164
F statistic: 3.2652 3, 16 degrees of freedom
. X2 X3 X4 Y
1: 7 52 8 -152
2: 39 20 42 1768
3: 5 61 44 -342
4: 32 74 37 878
5: 17 62 19 248
6: 8 69 13 -272
7: 22 5 33 1068
8: 11 42 8 148
9: 4 60 25 -382
10: 7 67 9 -302
11: 18 39 8 528
12: 25 51 30 758
13: 12 75 34 -132
14: 15 4 18 728
15: 22 58 2 538
16: 34 36 5 1358
17: 5 15 25 118
18: 22 75 35 368
19: 33 9 40 1578
20: 3 22 36 -52
r squared: 1.0000
Standard error: 8.0814896769E-09
Coefficient 1 X2 50.0000
t statistic: 291930436510.0000
Standard err: 0.0000
Coefficient 2 X3 -10.0000
t statistic: -129349508580.0000
Standard err: 0.0000
Coefficient 3 X4 0.0000
t statistic: 0.8296
Standard err: 0.0000
Coefficient 4 Constant 18.0000
t statistic: 2975240473.2000
Standard err: 0.0000
F statistic: 0.0000 3, 16 degrees of freedom
. X1 X2 X3 X4 Y
1: 5 7 52 8 -152
2: 39 39 20 42 1768
3: 36 5 61 44 -342
4: 47 32 74 37 878
5: 53 17 62 19 248
6: 99 8 69 13 -272
7: 67 22 5 33 1068
8: 90 11 42 8 148
9: 100 4 60 25 -382
10: 48 7 67 9 -302
11: 74 18 39 8 528
12: 43 25 51 30 758
13: 32 12 75 34 -132
14: 86 15 4 18 728
15: 36 22 58 2 538
16: 8 34 36 5 1358
17: 47 5 15 25 118
18: 96 22 75 35 368
19: 67 33 9 40 1578
20: 78 3 22 36 -52
r squared: 1.0000
Standard error: 1.0137673670E-08
Coefficient 1 X1 -0.0000
t statistic: -0.7027
Standard err: 0.0000
Coefficient 2 X2 50.0000
t statistic: 222166874780.0000
Standard err: 0.0000
Coefficient 3 X3 -10.0000
t statistic: -102374752610.0000
Standard err: 0.0000
Coefficient 4 X4 -0.0000
t statistic: -0.4268
Standard err: 0.0000
Coefficient 5 Constant 18.0000
t statistic: 1855905377.9000
Standard err: 0.0000
F statistic: 0.0000 4, 15 degrees of freedom
Problems 7 to 21 use the same values for the dependent variable Y and the
same set of independent variables X1, X2, X3, and X4.
Problem 14 illustrates that you can achieve a perfect fit
with independent variables X2 and X3. Problem 21 illustrates
that the other coefficients will have estimated coefficients of zero
if X2 and X3 are included. The other problems indicate what can happen
when different combinations of independent variables are used.
. X1 X2 X3 Y
1: 99 120 35 8708
2: 56 49 31 6355
3: 31 90 106 5970
4: 54 54 47 5354
5: 36 3 115 5557
6: 60 100 104 7272
7: 6 42 115 7262
8: 0 87 76 8623
9: 69 105 108 8207
10: 47 84 82 4500
11: 32 120 7 6220
12: 43 108 17 7642
13: 21 135 86 3875
14: 82 123 107 8335
15: 38 127 66 10009
16: 42 40 59 4216
17: 76 146 16 8980
18: 66 137 116 6325
19: 19 95 90 7843
20: 61 32 24 4314
r squared: 0.2372
Standard error: 1.6964385518E+03
Coefficient 1 X1 4.0427
t statistic: 0.2473
Standard err: 16.3472
Coefficient 2 X2 20.7441
t statistic: 2.0694
Standard err: 10.0241
Coefficient 3 X3 1.9095
t statistic: 0.1820
Standard err: 10.4920
Coefficient 4 Constant 4590.5502
t statistic: 3.2588
Standard err: 1408.6680
F statistic: 1.6580 3, 16 degrees of freedom
X1 X2 X4 Y
1: 99 120 74 8708
2: 56 49 22 6355
3: 31 90 71 5970
4: 54 54 68 5354
5: 36 3 49 5557
6: 60 100 46 7272
7: 6 42 3 7262
8: 0 87 79 8623
9: 69 105 34 8207
10: 47 84 50 4500
11: 32 120 1 6220
12: 43 108 45 7642
13: 21 135 17 3875
14: 82 123 63 8335
15: 38 127 71 10009
16: 42 40 27 4216
17: 76 146 25 8980
18: 66 137 24 6325
19: 19 95 32 7843
20: 61 32 63 4314
r squared: 0.3036
Standard error: 1.6208913125E+03
Coefficient 1 X1 -1.1288
t statistic: -0.0718
Standard err: 15.7172
Coefficient 2 X2 22.1208
t statistic: 2.2942
Standard err: 9.6421
Coefficient 3 X4 19.9152
t statistic: 1.2500
Standard err: 15.9321
Coefficient 4 Constant 3983.3948
t statistic: 3.3357
Standard err: 1194.1876
F statistic: 2.3249 3, 16 degrees of freedom
. X1 X3 X4 Y
1: 99 35 74 8708
2: 56 31 22 6355
3: 31 106 71 5970
4: 54 47 68 5354
5: 36 115 49 5557
6: 60 104 46 7272
7: 6 115 3 7262
8: 0 76 79 8623
9: 69 108 34 8207
10: 47 82 50 4500
11: 32 7 1 6220
12: 43 17 45 7642
13: 21 86 17 3875
14: 82 107 63 8335
15: 38 66 71 10009
16: 42 59 27 4216
17: 76 16 25 8980
18: 66 116 24 6325
19: 19 90 32 7843
20: 61 24 63 4314
r squared: 0.0746
Standard error: 1.8684387938E+03
Coefficient 1 X1 9.7396
t statistic: 0.5475
Standard err: 17.7904
Coefficient 2 X3 0.5486
t statistic: 0.0473
Standard err: 11.5921
Coefficient 3 X4 15.5295
t statistic: 0.8486
Standard err: 18.3001
Coefficient 4 Constant 5612.0956
t statistic: 3.8932
Standard err: 1441.5079
F statistic: 0.4301 3, 16 degrees of freedom
X2 X3 X4 Y
1: 120 35 74 8708
2: 49 31 22 6355
3: 90 106 71 5970
4: 54 47 68 5354
5: 3 115 49 5557
6: 100 104 46 7272
7: 42 115 3 7262
8: 87 76 79 8623
9: 105 108 34 8207
10: 84 82 50 4500
11: 120 7 1 6220
12: 108 17 45 7642
13: 135 86 17 3875
14: 123 107 63 8335
15: 127 66 71 10009
16: 40 59 27 4216
17: 146 16 25 8980
18: 137 116 24 6325
19: 95 90 32 7843
20: 32 24 63 4314
r squared: 0.3038
Standard error: 1.6206099548E+03
Coefficient 1 X2 21.9952
t statistic: 2.3804
Standard err: 9.2403
Coefficient 2 X3 1.0159
t statistic: 0.1035
Standard err: 9.8144
Coefficient 3 X4 19.6028
t statistic: 1.2646
Standard err: 15.5006
Coefficient 4 Constant 3883.7664
t statistic: 2.8401
Standard err: 1367.4764
F statistic: 2.3276 3, 16 degrees of freedom
. X1 X2 X3 X4 Y
1: 99 120 35 74 8708
2: 56 49 31 22 6355
3: 31 90 106 71 5970
4: 54 54 47 68 5354
5: 36 3 115 49 5557
6: 60 100 104 46 7272
7: 6 42 115 3 7262
8: 0 87 76 79 8623
9: 69 105 108 34 8207
10: 47 84 82 50 4500
11: 32 120 7 1 6220
12: 43 108 17 45 7642
13: 21 135 86 17 3875
14: 82 123 107 63 8335
15: 38 127 66 71 10009
16: 42 40 59 27 4216
17: 76 146 16 25 8980
18: 66 137 116 24 6325
19: 19 95 90 32 7843
20: 61 32 24 63 4314
r squared: 0.3039
Standard error: 1.6736253722E+03
D-W: 2.3080
Coefficient 1 X1 -0.8131
t statistic: -0.0489
Standard err: 16.6275
Coefficient 2 X2 22.1342
t statistic: 2.2230
Standard err: 9.9570
Coefficient 3 X3 0.9054
t statistic: 0.0872
Standard err: 10.3846
Coefficient 4 X4 19.7992
t statistic: 1.1997
Standard err: 16.5041
Coefficient 5 Constant 3908.7023
t statistic: 2.6033
Standard err: 1501.4595
F statistic: 1.6374 4, 15 degrees of freedom
X1 X2 X3 X4 Y
1: 99 120 35 74 132
2: 56 49 31 22 142
3: 31 90 106 71 66
4: 54 54 47 68 72
5: 36 3 115 49 106
6: 60 100 104 46 110
7: 6 42 115 3 186
8: 0 87 76 79 171
9: 69 105 108 34 142
10: 47 84 82 50 34
11: 32 120 7 1 132
12: 43 108 17 45 149
13: 21 135 86 17 19
14: 82 123 107 63 113
15: 38 127 66 71 190
16: 42 40 59 27 70
17: 76 146 16 25 175
18: 66 137 116 24 69
19: 19 95 90 32 161
20: 61 32 24 63 55
r squared: 0.0771
Standard error: 5.5787512408E+01
Coefficient 1 X1 -0.4271
t statistic: -0.7706
Standard err: 0.5543
Coefficient 2 X2 0.2378
t statistic: 0.7165
Standard err: 0.3319
Coefficient 3 X3 -0.2365
t statistic: -0.6832
Standard err: 0.3462
Coefficient 4 X4 -0.0067
t statistic: -0.0122
Standard err: 0.5501
Coefficient 5 Constant 130.2901
t statistic: 2.6033
Standard err: 50.0487
F statistic: 0.3135 4, 15 degrees of freedom
In exercise 27 each column was generated randomly, so there is no connection between
the independent variables and the dependent variable.
28.
Perform a regression calculation with T as the indepenent variable and Y as the
dependent variable. (In this case T represents time.) Then, calculate the
regression residuals and graph those values.
. T Y
1: 1 100.00000
2: 2 106.00000
3: 3 112.36000
4: 4 119.10160
5: 5 126.24770
6: 6 133.82256
7: 7 141.85191
8: 8 150.36303
9: 9 159.38481
10: 10 168.94790
11: 11 179.08477
12: 12 189.82986
13: 13 201.21965
14: 14 213.29283
15: 15 226.09040
16: 16 239.65582
17: 17 254.03517
18: 18 269.27728
19: 19 285.43392
20: 20 302.55995
r squared: 0.97829
Standard error: 9.4897383804E+00
Coefficient 1 T 10.48169
t statistic: 28.48315
Standard err: 0.36800
Coefficient 2 Constant 73.87017
t statistic: 16.75713
Standard err: 4.40828
F statistic: 811.28995 1, 18 degrees of freedom
There seems to be a good fit for this regression. The regression residuals
are:
1: 15.64813
2: 11.16644
3: 7.04475
4: 3.30465
5: -0.03094
6: -2.93778
7: -5.39012
8: -7.36070
9: -8.82061
10: -9.73921
11: -10.08403
12: -9.82064
13: -8.91254
14: -7.32106
15: -5.00518
16: -1.92145
17: 1.97620
18: 6.73662
19: 12.41156
20: 19.05590
The graph of these residuals do not show a random variation, but instead a clear
pattern: larger residuals at first, then smaller, then larger again. This clearly
indicates that this regression is not the best fit. In this case we have a curved
relationship, so we should perform another regression using the base-10 logarithm of Y
as the dependent variable:
. X Y LOG(Y)
1: 1 100.00000 2.00000
2: 2 106.00000 2.02531
3: 3 112.36000 2.05061
4: 4 119.10160 2.07592
5: 5 126.24770 2.10122
6: 6 133.82256 2.12653
7: 7 141.85191 2.15184
8: 8 150.36303 2.17714
9: 9 159.38481 2.20245
10: 10 168.94790 2.22775
11: 11 179.08477 2.25306
12: 12 189.82986 2.27836
13: 13 201.21965 2.30367
14: 14 213.29283 2.32898
15: 15 226.09040 2.35428
16: 16 239.65582 2.37959
17: 17 254.03517 2.40489
18: 18 269.27728 2.43020
19: 19 285.43392 2.45551
20: 20 302.55995 2.48081
r squared: 1.00000
Standard error: 6.8598386248E-12
Coefficient 1 X 0.02531
t statistic: 95130135393.00000
Standard err: 0.00000
Coefficient 2 Constant 1.97469
t statistic: 619684750750.00000
Standard err: 0.00000
F statistic: 0.00000 1, 18 degrees of freedom
Now there is a perfect fit.
This situation indicates constant percentage growth.