4. Consider the time series data in the file sunspot.dat. It consists of 285 observations of the number of sunspots from 1700 to 1984. This a quantity that is believed to affect our weather patterns. This time series has been studied by many authors like Yule etc. We will study the square root of the data (this transformation ensures that the variance is roughly constant). That is, for the Series Z₁, Z₂,... Zn from the file sunspot.dat, first compute the series Xt = √√Zt and work with the series {X} in what follows. (a) Compute the sample ACF and the sample PACF for this series (b) By considering the sample ACF and sample PACF, decide which of the following would be appropriate for this data AR(1), AR(2), MA (1) or MA(2). Use the data to estimate the parameters of the model you have chosen.
The 285 observations:
5.00
11.00
16.00
23.00
36.00
58.00
29.00
20.00
10.00
8.00
3.00
0.00
0.00
2.00
11.00
27.00
47.00
63.00
60.00
39.00
28.00
26.00
22.00
11.00
21.00
40.00
78.00
122.00
103.00
73.00
47.00
35.00
11.00
5.00
16.00
34.00
70.00
81.00
111.00
101.00
73.00
40.00
20.00
16.00
5.00
11.00
22.00
40.00
60.00
80.90
83.40
47.70
47.80
30.70
12.20
9.60
10.20
32.40
47.60
54.00
62.90
85.90
61.20
45.10
36.40
20.90
11.40
37.80
69.80
106.10
100.80
81.60
66.50
34.80
30.60
7.00
19.80
92.50
154.40
125.90
84.80
68.10
38.50
22.80
10.20
24.10
82.90
132.00
130.90
118.10
89.90
66.60
60.00
46.90
41.00
21.30
16.00
6.40
4.10
6.80
14.50
34.00
45.00
43.10
47.50
42.20
28.10
10.10
8.10
2.50
0.00
1.40
5.00
12.20
13.90
35.40
45.80
41.10
30.10
23.90
15.60
6.60
4.00
1.80
8.50
16.60
36.30
49.60
64.20
67.00
70.90
47.80
27.50
8.50
13.20
56.90
121.50
138.30
103.20
85.70
64.60
36.70
24.20
10.70
15.00
40.10
61.50
98.50
124.70
96.30
66.60
64.50
54.10
39.00
20.60
6.70
4.30
22.70
54.80
93.80
95.80
77.20
59.10
44.00
47.00
30.50
16.30
7.30
37.60
74.00
139.00
111.20
101.60
66.20
44.70
17.00
11.30
12.40
3.40
6.00
32.30
54.30
59.70
63.70
63.50
52.20
25.40
13.10
6.80
6.30
7.10
35.60
73.00
85.10
78.00
64.00
41.80
26.20
26.70
12.10
9.50
2.70
5.00
24.40
42.00
63.50
53.80
62.00
48.50
43.90
18.60
5.70
3.60
1.40
9.60
47.40
57.10
103.90
80.60
63.60
37.60
26.10
14.20
5.80
16.70
44.30
63.90
69.00
77.80
64.90
35.70
21.20
11.10
5.70
8.70
36.10
79.70
114.40
109.60
88.80
67.80
47.50
30.60
16.30
9.60
33.20
92.60
151.60
136.30
134.70
83.90
69.40
31.50
13.90
4.40
38.00
141.70
190.20
184.80
159.00
112.30
53.90
37.50
27.90
10.20
15.10
47.00
93.80
105.90
105.50
104.50
66.60
68.90
38.00
34.50
15.50
12.60
27.50
92.50
155.40
154.70
140.50
115.90
66.60
45.90
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