# The data file E2.txt on has two quarterly, seasonally adjusted U.S. investment series. From these, make the variables, assignment help

Anonymous

### Question Description

1. The data file E2.txt on has two quarterly, seasonally adjusted U.S. investment series. From these, make the variables

• ∇y1 = first differences of fixed investment

• ∇y2 = first differences of change in business inventories.

Use these data series from 1947-1968 only to answer the following questions:

(a) Plot the two series and comment on their stationarity.

(b) Test the hypothesis that ∇y2 does not Granger cause ∇y1.

(c) Estimate a VAR(p) for these data.

(d) Estimate the relevant impulse responses over 8 quarters. Report related measures of uncertainty and interepret the impulse responses.

### Unformatted Attachment Preview

y1 69.6 67.6 69.5 74.7 77.1 77.4 76.6 76.1 71.8 68.9 68.5 70.6 75.4 82.3 88.2 86.9 83.4 80.3 79.4 78.6 79.3 80.3 75.3 80.6 83.9 84.2 84.4 83.8 82.8 84.1 87.0 88.5 92.1 96.1 98.3 98.8 96.6 97.4 97.6 96.6 96.2 95.3 96.4 94.9 90.0 87.2 88.0 93.0 98.3 101.6 102.6 101.4 104.9 101.8 98.8 98.6 97.7 99.2 101.3 104.6 106.1 109.9 111.1 y2 0.1 -0.9 -2.9 2.7 4.1 5.6 6.9 5.3 -0.3 -7.1 -2.5 -7.7 4.4 7.7 8.0 22.1 13.4 19.9 14.6 7.0 7.3 -2.7 5.4 7.2 3.9 5.1 1.9 -5.0 -3.4 -4.1 -2.7 1.5 5.9 8.0 7.8 9.2 7.5 5.5 4.9 5.4 2.5 2.9 3.7 -3.0 -6.8 -6.2 0.3 5.3 5.0 13.0 -0.4 8.2 13.5 4.9 3.0 -3.9 -3.8 1.9 6.6 6.7 10.6 9.2 8.0 110.1 110.7 116.0 118.5 122.0 124.0 124.0 124.9 126.4 133.4 137.9 140.1 143.8 147.5 146.2 145.0 139.7 136.4 139.6 141.1 145.5 148.9 148.9 150.7 155.0 159.1 158.4 158.1 154.3 151.8 150.0 150.4 149.5 154.3 158.4 162.1 166.0 174.3 176.1 178.2 186.7 4.7 7.6 7.0 9.3 7.1 6.1 8.0 7.3 7.9 13.4 10.6 12.4 8.8 13.5 17.8 15.1 20.5 14.6 7.5 12.2 13.8 6.3 11.8 9.2 7.6 9.8 12.2 13.4 6.8 2.9 4.8 6.3 3.3 7.9 10.0 5.0 3.7 4.8 10.1 12.1 10.8 ...
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nancyW
School: Cornell University

Hi there!Attached is the complete solution to your question. This includes all of the statistical analysis that was done in Excel and then fully explained in Word.Thanks again!

a) Plots:
Time step / Box-Cox(y1)
5.1
5
4.9

Box-Cox(y1)

4.8
4.7
4.6
4.5
4.4
4.3
4.2
0

20

40

60

Time step

80

100

Time step / y2
25

20

15

y2

10

5

0
0

20

40

60

80

100

-5

-10

Time step

With regards to stationarity, we examine the Dickey-Fuller test for both series:
Dickey-Fuller test (ADF(stationary) / k: 4 / Box-Cox(y1)):

Tau (Observed-2.248
value)
Tau (Critical value)
-0.775
p-value (one-tailed)
0.439
alpha
0.05
Test interpretation:
H0: There is a unit root for the series.
Ha: There is no unit root for the series. The series is stationary.
As the computed p-value is greater than the significance level alpha=0.05, one cannot reject
the null hypothesis H0.
The risk to reject the null hypothesis H0 while it is true is 43.85%.
y1 fails the Dickey-Fuller test, and is therefore non-stationary.

Dickey-Fuller test (ADF(stationary) / k: 4 / y2):
Tau (Observed value)
Tau (Critical value)
p-value (one-tailed)
alpha

-3.419
-0.775
0.048
0.05

Test interpretation:
H0: There is a unit root for the series.
Ha: There is no unit root for the series. The series is stationary.
As the computed p-value is lower than the significance level alpha=0.05, one should reject the null
hypothesis H0, and accept the alternative hypothesis Ha.
The risk to reject the null hypothesis H0 while it is true is lower than 4.78%.

y2, however, has statistically significant evidence of stationarity. Therefore, y2 is the more
stationary data set.

b) As follows:
H0

Eigenvalue
None
0.158
At most 1
0.133
y1
y2

0.448
-1.259

Statistic
26.092
11.848

Critical value
20.262
9.164

p-value
0.007
0.015

0.699
0.988

Cointegration coefficients (beta):
y1
y2
Intercept

0.001
0.147
-0.014

0.045
-0.259
-2.379

We must reject the null hypothesis that y2 does not Granger-cause y1. Our results at the 0.05
level of significance reveal 1 cointegrating relationships between y2 and y1.

c) VAR(p) estimate:

VAR order estimation:
Number of lags

AIC
1
2
3
4
5

HQ
5.188
4.805
4.857
4.876
4.896

5.118
4.688
4.693
4.665
4.638

BIC
5.293
4.979
5.101
5.190
5.280

The VAR order estimate according to AIC is 5.

d) Impulse responses:
ARIMA (y1)
180
160

y1

140
120
100
80
60
0

20

40

60

80

Time step
y1

ARIMA (y1)

Validation

Prediction

Lower bound (95%)

Upper bound (95%)

100

FPE
167.021
108.657
109.300
106.388
103.711

ARIMA (y2)
25
20
15

y2

10
5
0
0

20

40

60

80

-5
-10

Time step
y2

ARIMA (y2)

Validation

Prediction

Lower bound (95%)

Upper bound (95%)

100

Date

y1
1947
1947
1947
1947
1948
1948
1948
1948
1949
1949
1949
1949
1950
1950
1950
1950
1951
1951
1951
1951
1952
1952
1952
1952
1953
1953
1953
1953
1954
1954
1954
1954
1955
1955
1955
1955
1956
1956
1956
1956
1957
1957
1957
1957
1958
1958

y2
69,6
67,6
69,5
74,7
77,1
77,4
76,6
76,1
71,8
68,9
68,5
70,6
75,4
82,3
88,2
86,9
83,4
80,3
79,4
78,6
79,3
80,3
75,3
80,6
83,9
84,2
84,4
83,8
82,8
84,1
87
88,5
92,1
96,1
98,3
98,8
96,6
97,4
97,6
96,6
96,2
95,3
96,4
94,9
90
87,2

Quarter
0,1
-0,9
-2,9
2,7
4,1
5,6
6,9
5,3
-0,3
-7,1
-2,5
-7,7
4,4
7,7
8
22,1
13,4
19,9
14,6
7
7,3
-2,7
5,4
7,2
3,9
5,1
1,9
-5
-3,4
-4,1
-2,7
1,5
5,9
8
7,8
9,2
7,5
5,5
4,9
5,4
2,5
2,9
3,7
-3
-6,8
-6,2

y1
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
33
34
35
36
37
38
39
40
41
42
43
44
45
46

y2
69,6
67,6
69,5
74,7
77,1
77,4
76,6
76,1
71,8
68,9
68,5
70,6
75,4
82,3
88,2
86,9
83,4
80,3
79,4
78,6
79,3
80,3
75,3
80,6
83,9
84,2
84,4
83,8
82,8
84,1
87
88,5
92,1
96,1
98,3
98,8
96,6
97,4
97,6
96,6
96,2
95,3
96,4
94,9
90
87,2

0,1
-0,9
-2,9
2,7
4,1
5,6
6,9
5,3
-0,3
-7,1
-2,5
-7,7
4,4
7,7
8
22,1
13,4
19,9
14,6
7
7,3
-2,7
5,4
7,2
3,9
5,1
1,9
-5
-3,4
-4,1
-2,7
1,5
5,9
8
7,8
9,2
7,5
5,5
4,9
5,4
2,5
2,9
3,7
-3
-6,8
-6,2

1958
1958
1959
1959
1959
1959
1960
1960
1960
1960
1961
1961
1961
1961
1962
1962
1962
1962
1963
1963
1963
1963
1964
1964
1964
1964
1965
1965
1965
1965
1966
1966
1966
1966
1967
1967
1967
1967
1968
1968
1968
1968

88
93
98,3
101,6
102,6
101,4
104,9
101,8
98,8
98,6
97,7
99,2
101,3
104,6
106,1
109,9
111,1
110,1
110,7
116
118,5
122
124
124
124,9
126,4
133,4
137,9
140,1
143,8
147,5
146,2
145
139,7
136,4
139,6
141,1
145,5
148,9
148,9
150,7
155

0,3
5,3
5
13
-0,4
8,2
13,5
4,9
3
-3,9
-3,8
1,9
6,6
6,7
10,6
9,2
8
4,7
7,6
7
9,3
7,1
6,1
8
7,3
7,9
13,4
10,6
12,4
8,8
13,5
17,8
15,1
20,5
14,6
7,5
12,2
13,8
6,3
11,8
9,2
7,6

47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88

88
93
98,3
101,6
102,6
101,4
104,9
101,8
98,8
98,6
97,7
99,2
101,3
104,6
106,1
109,9
111,1
110,1
110,7
116
118,5
122
124
124
124,9
126,4
133,4
137,9
140,1
143,8
147,5
146,2
145
139,7
136,4
139,6
141,1
145,5
148,9
148,9
150,7
155

0,3
5,3
5
13
-0,4
8,2
13,5
4,9
3
-3,9
-3,8
1,9
6,6
6,7
10,6
9,2
8
4,7
7,6
7
9,3
7,1
6,1
8
7,3
7,9
13,4
10,6
12,4
8,8
13,5
17,8
15,1
20,5
14,6
7,5
12,2
13,8
6,3
11,8
9,2
7,6

XLSTAT 2016.07.39045 - ARIMA - Start time: 12/11/2016 at 11:29:02 AM / End time: 12/11/2016 at 11:29:03 AM / Mi
Time series: Workbook = time series analysis.xlsx / Sheet = Sheet1 / Range = Sheet1!\$F\$1:\$G\$89 / 88 rows and 2 colum
Confidence intervals (%): 95
Center: No
Model parameters: p = 1 / d = 0 / q = 0 / P = 0 / D = 0 / Q = 0 / s = 0
Optimize: Likelihood (Convergence = 0.00001 / Iterations ...

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