1.
The table given below gives data on expenditure on food and total expenditure, measured in rupees, for a sample of
55 rural households from India. (In year 2000, a U.S. dollar was about 40 Indian rupees.)
(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.
observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
x
Food expenditure
217
196
303
270
325
260
300
325
336
345
325
362
315
355
325
325
370
390
420
410
383
315
y
Total expenditure
382
388
391
415
456
460
472
478
494
516
525
554
575
579
585
586
590
608
610
616
618
623
x
y
Food expenditure
Total expenditure
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
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
480
352
410
380
610
530
360
627
630
635
640
648
650
655
662
663
677
680
690
695
695
695
720
721
730
731
733
745
751
752
752
769
773
773
775
785
788
790
795
55
305
20449
801
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
Regression
Residual
Total
df
1
53
54
SS
272893.6099
455730.3174
728623.9273
MS
272893.61
8598.68523
Ftab
0.00396886
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
Intercept
Food
expenditure
Coefficients
322.6171555
Standard Error
57.54203228
t Stat
5.60663471
P-value
7.59298E-07
Lower 95%
207.2024547
Upper 95%
438.0319
Lower
95.0%
207.2025
Upper
95.0%
438.0319
0.851046821
0.15106811
5.6335306
6.88792E-07
0.548042564
1.154051
0.548043
1.154051
Ttab =
2.005746
Testing the hypothesis for intercept
H0: β0 = 0
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
1
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
Predicted Total
expenditure
507.2943157
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
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
Residuals
250
200
150
100
50
0
-50 0
100
200
300
400
500
600
700
800
900
-100
-150
-200
-250
From the above graph we can say that the data contain heterosce
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
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
-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
(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
1
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
Predicted Total
expenditure
507.2943157
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
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
ui2
15698.6656
10286.4895
35904.316
18878.7043
20508.3515
7037.41953
11221.4195
14691.2271
13126.03
10045.7139
5506.73412
5882.29249
246.392801
2092.03573
201.849431
174.434686
2256.67556
2164.61431
4907.95809
3085.39724
934.408006
1043.49
5952.63837
2505.68527
3256.84772
16939.6847
309.373781
871.221238
0.22523018
137.588638
ln(ui2)
9.661330991
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
-1.49063237
4.924268349
ln(X)
5.379897354
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
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
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
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.304826638
R Square
0.092919279
-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
3553.25213
6455.44244
10441.2508
1.25536134
14719.5208
6893.47032
2823.91977
207.699257
2042.8966
2723.00693
10405.2218
751.405406
7433.77633
3897.84345
21560.6053
8375.40823
4464.37281
1753.96542
22744.9722
10702.6572
19316.8448
2889.67705
266.60454
27557.9883
47879.3758
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
9.868732782
7.968900027
5.585766436
10.22404773
10.77644012
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
5.940171253
6.413458957
6.272877007
5.886104031
5.720311777
Adjusted R
Square
Standard Error
Observations
0.075804549
2.157486297
55
ANOVA
Regression
Residual
Total
df
1
53
54
SS
25.27154869
246.7015976
271.9731462
MS
25.2715487
4.65474712
F
5.429199056
Significance F
0.023643036
Intercept
ln(X)
Coefficients
25.62947103
-2.997878479
Standard Error
7.587948891
1.286607688
t Stat
3.37765467
-2.3300642
P-value
0.001376395
0.023643036
Lower 95%
10.40997292
-5.578486697
ttab =
2.005746
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
Upper 95%
40.84897
-0.41727
Lower
95.0%
10.40997
-5.57849
Upper
95.0%
40.84897
-0.41727
Observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Predicted Total
expenditure
507.2943157
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
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
abs(ui )
125.294316
101.422332
189.484342
137.399797
143.207372
83.889329
105.931202
121.207372
114.568887
100.228309
74.2073724
76.6961048
15.6969042
45.738777
14.2073724
13.2073724
47.5044794
46.5254158
70.0568204
55.5463522
30.568088
32.3030958
77.1533432
50.0568204
57.0687981
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
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
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
130.152544
17.5890245
29.5164571
0.47458422
11.7298183
59.6091615
80.3457681
102.182439
1.12042909
121.324032
83.0269253
53.1405661
14.411775
45.1984137
52.1824389
102.005989
27.411775
86.2193501
62.4327114
146.8353
91.5172565
66.8159622
41.8803703
150.814363
103.453648
138.985052
53.7557164
16.3280293
166.005989
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
Regression
Residual
Total
Intercept
Food
expenditure
We will test
df
1
53
54
SS
22951.76191
111371.3124
134323.0743
MS
22951.7619
2101.34552
F
10.9224122
Significance F
0.001708262
Coefficients
168.2089301
Standard Error
28.44578378
t Stat
5.91331676
P-value
2.48665E-07
Lower 95%
111.1539132
Upper 95%
225.2639
Lower
95.0%
111.1539
Upper
95.0%
225.2639
-0.246811156
0.074680206
-3.3049073
0.001708262
-0.396600679
-0.09702
-0.3966
-0.09702
ttab
2.005746
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.
ui2 = α0 + α1X + α2X2
In this, we'll fit the model of the form :
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
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
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
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
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
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
19316.8448
2889.677048
266.6045397
27557.98833
47879.3758
300
220
403
350
390
385
470
322
540
433
295
340
500
450
415
540
360
450
395
430
332
397
446
480
352
410
380
610
530
360
305
90000
48400
162409
122500
152100
148225
220900
103684
291600
187489
87025
115600
250000
202500
172225
291600
129600
202500
156025
184900
110224
157609
198916
230400
123904
168100
144400
372100
280900
129600
93025
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.401568399
0.161257179
0.12899784
8990.618725
55
ANOVA
Regression
Residual
Total
df
2
52
54
SS
808114215
4203223703
5011337919
Intercept
X
x^2
Coefficients
36525.9491
-106.4959758
0.078265849
Standard
Error
18350.0769
95.0257685
0.12011485
MS
404057107.7
80831225.07
F
4.998775008
Significance
F
0.010336
t Stat
1.990506596
-1.12070628
0.651591782
P-value
0.051803225
0.267562148
0.517534135
Lower 95%
-296.174
-297.179
-0.16276
We will test
H0: Data does not involves hetroscedasticity
H1: Data involves hetroscedasticity
R2
n
0.161
55
nR2
8.855
Test statistic:
nR2 ~ χ2df,α
Upper
95%
73348.07
84.18718
0.319294
Lower
95.0%
-296.174
-297.179
-0.16276
Up
95.
733
84.
0.3
where, df = no. of independent variables in Auxilliary Model.
Here, df = 2
χ2cal = χ2df,0.05
χ2df,0.05 =
5.99146455
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