1. The table given below gives data on expenditure on food and total expenditure, measured in rupees, for a sample of 55 r
(i) Fit Regression model to food expenditure on total expenditure and interpret the results
. (ii) Use the Park, Glejser, and White test to find out if the impression of heteroscedasticity is supported by these tests.
x
y
observation Food expenditure Total expenditure
1
217
382
2
196
388
3
303
391
4
270
415
5
325
456
6
260
460
7
300
472
8
325
478
9
336
494
10
345
516
11
325
525
12
362
554
13
315
575
14
355
579
15
325
585
16
325
586
17
370
590
18
390
608
19
420
610
20
410
616
21
383
618
22
315
623
23
267
627
24
420
630
25
300
635
26
220
640
27
403
648
28
350
650
29
390
655
30
385
662
31
470
663
32
322
677
33
540
680
34
433
690
35
295
695
36
340
695
37
500
695
38
450
720
39
415
721
40
540
730
x
y
Food expenditure
Total expenditure
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
360
450
395
430
332
397
446
480
352
410
380
610
530
360
305
731
733
745
751
752
752
769
773
773
775
785
788
790
795
801
20449
35147
(i) Fit Regression model to food expenditure on total expenditure and interpret the results.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.611990906
R Square
0.374532869
Adjusted R Square 0.362731602
Standard Error
92.72909594
Observations
55
Here, we will test the hypothesis
H0: Regression model is not significant .
H0: Regression model is significant .
ANOVA
df
Regression
Residual
Total
SS
MS
272893.6099 272893.6099
455730.3174 8598.685234
728623.9273
1
53
54
Ftab
0.003968861
Here , Fcal > Ftab So , we reject H0 at 5% level of significance
Hence , Regression model is significant
F
31.736667
Significance F
6.88792E-07
Coefficients
Intercept
322.6171555
Food expenditure 0.851046821
Standard Error
t Stat
57.54203228 5.606634712
0.15106811 5.633530598
Ttab =
P-value
7.59298E-07
6.88792E-07
Lower 95%
207.2024547
0.548042564
2.005745995
H0: β0 = 0
Testing the hypothesis for intercept
H1: β0 ≠ 0
Here, tcal > ttab. So we reject Ho: st 5% los
Hence β0 is significant
Testing the hypothesis for food expenditure (β1)
H0: β1 = 0
H1: β1 ≠ 0
Here, tcal > ttab. So we reject Ho: st 5% los
Hence β1 or SP affect the car mileage.
Hence the fitted line is yᶺ = 322.61 + 0.85 food expenditure
RESIDUAL OUTPUT
Observation
Predicted Total expenditure
1
507.2943157
2
489.4223325
3
580.4843423
4
552.3997972
5
599.2073724
6
543.889329
7
577.9312019
8
599.2073724
9
608.5688874
10
616.2283088
11
599.2073724
12
630.6961048
13
590.6969042
14
624.738777
15
599.2073724
16
599.2073724
17
637.5044794
18
654.5254158
Residuals
-125.2943157
-101.4223325
-189.4843423
-137.3997972
-143.2073724
-83.88932904
-105.9312019
-121.2073724
-114.5688874
-100.2283088
-74.20737241
-76.69610479
-15.6969042
-45.73877704
-14.20737241
-13.20737241
-47.50447936
-46.52541578
250
200
150
100
50
0
-50 0
-100
-150
-200
-250
100
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
680.0568204
671.5463522
648.568088
590.6969042
549.8466568
680.0568204
577.9312019
509.8474562
665.5890245
620.4835429
654.5254158
650.2701817
722.6091615
596.6542319
782.1824389
691.1204291
573.6759678
611.9730747
748.1405661
705.588225
675.8015863
782.1824389
628.9940111
705.588225
658.7806499
688.5672886
605.1647002
660.4827435
702.1840378
731.1196297
622.1856366
671.5463522
646.0149476
841.7557164
773.6719707
628.9940111
582.186436
-70.05682041
-55.5463522
-30.56808803
32.3030958
77.15334322
-50.05682041
57.06879812
130.1525438
-17.58902445
29.51645706
0.47458422
11.72981833
-59.60916147
80.34576805
-102.1824389
-1.120429088
121.3240322
83.02692527
-53.1405661
14.41177495
45.19841369
-52.18243895
102.0059889
27.41177495
86.21935011
62.43271138
146.8352998
91.51725647
66.81596224
41.88037032
150.8143634
103.4536478
138.9850524
-53.75571642
16.32802926
166.0059889
218.813564
From the above graph we can say that
(ii) Use the Park, Glejser, and White test to find out if the impression of heteroscedasticity is supported by these tests.
1 . Park test
In this, we'll fit the model of the form : lnu i2 = α + βlnXi + vi , i = 1,2,….n
Observation
Predicted Total expenditure
1
507.2943157
ui2
Residuals
-125.2943157 15698.66555
ln(ui2)
9.661330991
ln(X)
5.379897354
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
489.4223325
580.4843423
552.3997972
599.2073724
543.889329
577.9312019
599.2073724
608.5688874
616.2283088
599.2073724
630.6961048
590.6969042
624.738777
599.2073724
599.2073724
637.5044794
654.5254158
680.0568204
671.5463522
648.568088
590.6969042
549.8466568
680.0568204
577.9312019
509.8474562
665.5890245
620.4835429
654.5254158
650.2701817
722.6091615
596.6542319
782.1824389
691.1204291
573.6759678
611.9730747
748.1405661
705.588225
675.8015863
782.1824389
628.9940111
705.588225
658.7806499
688.5672886
605.1647002
660.4827435
702.1840378
731.1196297
622.1856366
671.5463522
-101.4223325
-189.4843423
-137.3997972
-143.2073724
-83.88932904
-105.9312019
-121.2073724
-114.5688874
-100.2283088
-74.20737241
-76.69610479
-15.6969042
-45.73877704
-14.20737241
-13.20737241
-47.50447936
-46.52541578
-70.05682041
-55.5463522
-30.56808803
32.3030958
77.15334322
-50.05682041
57.06879812
130.1525438
-17.58902445
29.51645706
0.47458422
11.72981833
-59.60916147
80.34576805
-102.1824389
-1.120429088
121.3240322
83.02692527
-53.1405661
14.41177495
45.19841369
-52.18243895
102.0059889
27.41177495
86.21935011
62.43271138
146.8352998
91.51725647
66.81596224
41.88037032
150.8143634
103.4536478
10286.48953
35904.31599
18878.70428
20508.35151
7037.419526
11221.41953
14691.22713
13126.02997
10045.71389
5506.73412
5882.29249
246.3928014
2092.035725
201.8494308
174.4346859
2256.675559
2164.614314
4907.958086
3085.397243
934.408006
1043.489998
5952.638369
2505.68527
3256.847719
16939.68466
309.3737813
871.2212376
0.225230181
137.5886379
3553.252131
6455.442444
10441.25083
1.255361341
14719.5208
6893.470321
2823.919766
207.6992573
2042.8966
2723.006934
10405.22176
751.4054061
7433.776334
3897.84345
21560.60528
8375.408232
4464.37281
1753.965418
22744.97221
10702.65724
9.238586617
10.48861279
9.845789808
9.928587473
8.858996837
9.325579689
9.5950058
9.482352558
9.214901344
8.613727008
8.679701844
5.506927016
7.645892902
5.307522027
5.161550379
7.721648018
7.679997478
8.498613266
8.034435694
6.83991318
6.950326142
8.691589824
7.826317537
8.088515049
9.737414353
5.734550194
6.769895949
-1.490632371
4.924268349
8.175618556
8.772678844
9.253519665
0.227423452
9.596929837
8.838329912
7.945881187
5.336091155
7.622123982
7.909492039
9.250063052
6.621945328
8.913789264
8.268178718
9.978623099
9.0330551
8.403884015
7.469634457
10.0320994
9.27824733
5.278114659
5.713732806
5.598421959
5.783825182
5.560681631
5.703782475
5.783825182
5.81711116
5.843544417
5.783825182
5.891644212
5.752572639
5.872117789
5.783825182
5.783825182
5.913503006
5.966146739
6.040254711
6.01615716
5.948034989
5.752572639
5.587248658
6.040254711
5.703782475
5.393627546
5.998936562
5.857933154
5.966146739
5.953243334
6.152732695
5.774551546
6.29156914
6.070737728
5.686975356
5.828945618
6.214608098
6.109247583
6.02827852
6.29156914
5.886104031
6.109247583
5.978885765
6.063785209
5.805134969
5.983936281
6.100318952
6.173786104
5.863631176
6.01615716
51
52
53
54
55
646.0149476
841.7557164
773.6719707
628.9940111
582.186436
138.9850524 19316.8448
-53.75571642 2889.677048
16.32802926 266.6045397
166.0059889 27557.98833
218.813564 47879.3758
9.868732782
7.968900027
5.585766436
10.22404773
10.77644012
5.940171253
6.413458957
6.272877007
5.886104031
5.720311777
1
53
54
SS
MS
25.27154869 25.27154869
246.7015976 4.654747124
271.9731462
F
5.429199056
Significance F
0.023643036
Coefficients
25.62947103
-2.997878479
Standard Error
t Stat
7.587948891 3.377654672
1.286607688 -2.330064174
P-value
0.001376395
0.023643036
Lower 95%
10.40997292
-5.578486697
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.304826638
R Square
0.092919279
Adjusted R Square 0.075804549
Standard Error
2.157486297
Observations
55
ANOVA
df
Regression
Residual
Total
Intercept
ln(X)
ttab =
2.005745995
We will test H0: Data does not involves hetroscedasticity
H1: Data involves hetroscedasticity
Here P value < 0.05 So we reject Ho
Here, |tcal| . ttab. So we reject Ho: at 5% los
The park test confirm that the data involves heteroscedasticity.
2)
Glejser test
In this we will fit the model of the form:
|ui| = β1 + β2X + vi
Observation
Predicted Total expenditure
1
507.2943157
2
489.4223325
3
580.4843423
4
552.3997972
5
599.2073724
6
543.889329
7
577.9312019
8
599.2073724
9
608.5688874
10
616.2283088
11
599.2073724
12
630.6961048
13
590.6969042
14
624.738777
15
599.2073724
16
599.2073724
17
637.5044794
18
654.5254158
19
680.0568204
20
671.5463522
21
648.568088
22
590.6969042
23
549.8466568
24
680.0568204
25
577.9312019
26
509.8474562
27
665.5890245
28
620.4835429
29
654.5254158
30
650.2701817
31
722.6091615
32
596.6542319
33
782.1824389
34
691.1204291
35
573.6759678
36
611.9730747
37
748.1405661
38
705.588225
39
675.8015863
40
782.1824389
41
628.9940111
42
705.588225
43
658.7806499
44
688.5672886
45
605.1647002
46
660.4827435
Residuals
-125.2943157
-101.4223325
-189.4843423
-137.3997972
-143.2073724
-83.88932904
-105.9312019
-121.2073724
-114.5688874
-100.2283088
-74.20737241
-76.69610479
-15.6969042
-45.73877704
-14.20737241
-13.20737241
-47.50447936
-46.52541578
-70.05682041
-55.5463522
-30.56808803
32.3030958
77.15334322
-50.05682041
57.06879812
130.1525438
-17.58902445
29.51645706
0.47458422
11.72981833
-59.60916147
80.34576805
-102.1824389
-1.120429088
121.3240322
83.02692527
-53.1405661
14.41177495
45.19841369
-52.18243895
102.0059889
27.41177495
86.21935011
62.43271138
146.8352998
91.51725647
abs(u i )
125.2943157
101.4223325
189.4843423
137.3997972
143.2073724
83.88932904
105.9312019
121.2073724
114.5688874
100.2283088
74.20737241
76.69610479
15.6969042
45.73877704
14.20737241
13.20737241
47.50447936
46.52541578
70.05682041
55.5463522
30.56808803
32.3030958
77.15334322
50.05682041
57.06879812
130.1525438
17.58902445
29.51645706
0.47458422
11.72981833
59.60916147
80.34576805
102.1824389
1.120429088
121.3240322
83.02692527
53.1405661
14.41177495
45.19841369
52.18243895
102.0059889
27.41177495
86.21935011
62.43271138
146.8352998
91.51725647
47
48
49
50
51
52
53
54
55
702.1840378
731.1196297
622.1856366
671.5463522
646.0149476
841.7557164
773.6719707
628.9940111
582.186436
66.81596224
41.88037032
150.8143634
103.4536478
138.9850524
-53.75571642
16.32802926
166.0059889
218.813564
66.81596224
41.88037032
150.8143634
103.4536478
138.9850524
53.75571642
16.32802926
166.0059889
218.813564
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.41336405
R Square
0.170869838
Adjusted R Square 0.155225872
Standard Error
45.8404354
Observations
55
ANOVA
df
1
53
54
SS
MS
22951.76191 22951.76191
111371.3124 2101.345517
134323.0743
F
10.9224122
Significance F
0.001708262
Coefficients
Intercept
168.2089301
Food expenditure -0.246811156
Standard Error
t Stat
28.44578378 5.913316763
0.074680206 -3.30490729
P-value
2.48665E-07
0.001708262
Lower 95%
111.1539132
-0.396600679
Regression
Residual
Total
ttab
2.005745995
We will test H0: Data does not involves hetroscedasticity
H1: Data involves hetroscedasticity
Here, P value < 0.05 So we reject Ho
Here, | tcal | > ttab. So we reject Ho: at 5% los
Therefore, the data involves heterodcedasticity.
3 White's General Heteroscedasticity test.
In this, we'll fit the model of the form :
ui2 = α0 + α1X + α2X2
Residuals
-125.2943157
-101.4223325
-189.4843423
-137.3997972
-143.2073724
-83.88932904
-105.9312019
-121.2073724
-114.5688874
-100.2283088
-74.20737241
-76.69610479
-15.6969042
-45.73877704
-14.20737241
-13.20737241
-47.50447936
-46.52541578
-70.05682041
-55.5463522
-30.56808803
32.3030958
77.15334322
-50.05682041
57.06879812
130.1525438
-17.58902445
29.51645706
0.47458422
11.72981833
-59.60916147
80.34576805
-102.1824389
-1.120429088
121.3240322
83.02692527
-53.1405661
14.41177495
45.19841369
-52.18243895
102.0059889
27.41177495
86.21935011
62.43271138
146.8352998
91.51725647
66.81596224
ui2
15698.66555
10286.48953
35904.31599
18878.70428
20508.35151
7037.419526
11221.41953
14691.22713
13126.02997
10045.71389
5506.73412
5882.29249
246.3928014
2092.035725
201.8494308
174.4346859
2256.675559
2164.614314
4907.958086
3085.397243
934.408006
1043.489998
5952.638369
2505.68527
3256.847719
16939.68466
309.3737813
871.2212376
0.225230181
137.5886379
3553.252131
6455.442444
10441.25083
1.255361341
14719.5208
6893.470321
2823.919766
207.6992573
2042.8966
2723.006934
10405.22176
751.4054061
7433.776334
3897.84345
21560.60528
8375.408232
4464.37281
X
217
196
303
270
325
260
300
325
336
345
325
362
315
355
325
325
370
390
420
410
383
315
267
420
300
220
403
350
390
385
470
322
540
433
295
340
500
450
415
540
360
450
395
430
332
397
446
x^2
47089
38416
91809
72900
105625
67600
90000
105625
112896
119025
105625
131044
99225
126025
105625
105625
136900
152100
176400
168100
146689
99225
71289
176400
90000
48400
162409
122500
152100
148225
220900
103684
291600
187489
87025
115600
250000
202500
172225
291600
129600
202500
156025
184900
110224
157609
198916
41.88037032
150.8143634
103.4536478
138.9850524
-53.75571642
16.32802926
166.0059889
218.813564
1753.965418
22744.97221
10702.65724
19316.8448
2889.677048
266.6045397
27557.98833
47879.3758
480
352
410
380
610
530
360
305
230400
123904
168100
144400
372100
280900
129600
93025
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.401568399
R Square
0.161257179
Adjusted R Square
0.12899784
Standard Error
8990.618725
Observations
55
ANOVA
df
SS
2 808114215.5
52 4203223703
54 5011337919
MS
404057107.7
80831225.07
F
4.998775008
Coefficients
Standard Error
36525.9491 18350.0769
-106.4959758 95.02576853
0.078265849 0.12011485
t Stat
1.990506596
-1.120706282
0.651591782
P-value
0.051803225
0.267562148
0.517534135
Regression
Residual
Total
Intercept
X
x^2
We will test
R2
n
nR2
Test statistic:
H0: Data does not involves hetroscedasticity
H1: Data involves hetroscedasticity
0.161
55
8.855
nR2 ~ χ2df,α
where, df = no. of independent variables in Auxilliary Model.
Here, df = 2
χ2cal = χ2df,0.05
χ2df,0.05 =
5.991464547
So, nR2 > χ2df,0.05
So, we reject H0 at 5% level of significance
Hence data involves heteroscedasticity according to White's Test
Therefore the park,Glejser, and White's test concludes that the data invovles heteroscedasticity
n rupees, for a sample of 55 rural households from India. (In year 2000, a U.S. dollar was about 40 Indian rupees.)
Upper 95% Lower 95.0%Upper 95.0%
438.03186 207.20245 438.03186
1.1540511 0.5480426 1.1540511
Residuals
100
200
300
400
500
600
700
800
900
Upper 95% Lower 95.0%Upper 95.0%
40.848969 10.409973 40.848969
-0.41727 -5.578487
-0.41727
Upper 95% Lower 95.0%Upper 95.0%
225.26395 111.15391 225.26395
-0.097022 -0.396601 -0.097022
Significance F
0.0103359
Lower 95%
-296.1741
-297.1791
-0.162762
Upper 95% Lower 95.0%Upper 95.0%
73348.072 -296.1741 73348.072
84.187179 -297.1791 84.187179
0.3192939 -0.162762 0.3192939