Copper example
June 18, 2021
1 (a) Calculating the log returns of Copper
[9]: !pip install pandas
Requirement already satisfied: pandas in c:\users\rluck\anaconda3\lib\site-
packages (1.2.4)
Requirement already satisfied: pytz>=2017.3 in
c:\users\rluck\anaconda3\lib\site-packages (from pandas) (2021.1)
Requirement already satisfied: numpy>=1.16.5 in
c:\users\rluck\anaconda3\lib\site-packages (from pandas) (1.20.1)
Requirement already satisfied: python-dateutil>=2.7.3 in
c:\users\rluck\anaconda3\lib\site-packages (from pandas) (2.8.1)
Requirement already satisfied: six>=1.5 in c:\users\rluck\anaconda3\lib\site-
packages (from python-dateutil>=2.7.3->pandas) (1.15.0)
[10]: import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import statsmodels.api as sm
[11]: #Reading the data from Excel
data = pd.read_csv(“C:\\Users\\rluck\\OneDrive\\commodity.csv”)
data
[11]: OBS COPPER GOLD LEAD SILVER
0 5/01/1989 3039.1050 363.6 4823.0596 5.091
1 5/02/1989 2976.1080 361.8 4696.2984 5.110
2 5/03/1989 2924.9100 359.9 4592.1087 5.045
3 5/04/1989 3005.0100 360.6 4678.8006 5.076
4 5/05/1989 2969.1990 360.6 4623.0428 5.076
.. … … … … …
995 2/22/1993 1872.8360 368.6 2766.1788 4.256
996 2/23/1993 1872.1830 374.5 2757.7256 4.392
997 2/24/1993 1873.7095 375.1 2752.4793 4.389
998 2/25/1993 1883.5065 377.2 2785.7061 4.489
999 2/26/1993 1884.7230 375.2 2821.4303 4.448
[1000 rows x 5 columns]
1
#R = ln(P_t/P_t-1)
[12]: data[‘R_c’]= 100*np.log(data[‘COPPER’]/data[‘COPPER’].shift(1))
data[‘R_ca’]= abs(data[‘R_c’])
data[‘R_c^2’] = data[‘R_c’]*data[‘R_c’]
[13]: print(data.head())
OBS COPPER GOLD LEAD SILVER R_c R_ca R_c^2
0 5/01/1989 3039.105 363.6 4823.0596 5.091 NaN NaN NaN
1 5/02/1989 2976.108 361.8 4696.2984 5.110 -2.094666 2.094666 4.387625
2 5/03/1989 2924.910 359.9 4592.1087 5.045 -1.735270 1.735270 3.011160
3 5/04/1989 3005.010 360.6 4678.8006 5.076 2.701718 2.701718 7.299283
4 5/05/1989 2969.199 360.6 4623.0428 5.076 -1.198868 1.198868 1.437284
[14]: # Dropping NA’s is required to use numpy’s polyfit
data_s = data.dropna(subset=[‘R_c’])
print(data_s.head())
OBS COPPER GOLD LEAD SILVER R_c R_ca R_c^2
1 5/02/1989 2976.108 361.8 4696.2984 5.110 -2.094666 2.094666 4.387625
2 5/03/1989 2924.910 359.9 4592.1087 5.045 -1.735270 1.735270 3.011160
3 5/04/1989 3005.010 360.6 4678.8006 5.076 2.701718 2.701718 7.299283
4 5/05/1989 2969.199 360.6 4623.0428 5.076 -1.198868 1.198868 1.437284
5 5/08/1989 2926.000 359.6 4506.0400 5.067 -1.465592 1.465592 2.147959
2 (b) (i) Summary statistics
[15]: !pip install scipy
Requirement already satisfied: scipy in c:\users\rluck\anaconda3\lib\site-
packages (1.6.2)
Requirement already satisfied: numpy<1.23.0,>=1.16.5 in
c:\users\rluck\anaconda3\lib\site-packages (from scipy) (1.20.1)
[16]: from scipy import stats
Rc= stats.describe(data_s[‘R_c’])
Rca=stats.describe(data_s[‘R_ca’])
Rc2= stats.describe(data_s[‘R_c^2’])
[17]: print(Rc)
DescribeResult(nobs=999, minmax=(-10.037918530762216, 6.633374744570145),
mean=-0.04782604641413218, variance=2.988453995572702,
skewness=-0.1364205630955574, kurtosis=2.6804296191644488)
[18]: print(Rca)
2
DescribeResult(nobs=999, minmax=(0.0, 10.037918530762216),
mean=1.253037876686897, variance=1.4190664473695909,
skewness=1.9654561151759231, kurtosis=6.1340532142037)
[19]: print(Rc2)
DescribeResult(nobs=999, minmax=(0.0, 100.75980843021948),
mean=2.987749880847295, variance=41.92050785896554, skewness=6.482381876231294,
kurtosis=67.57368954840663)
3 b(iii) Jacques-Bera test
[20]: stats.jarque_bera(data_s[‘R_c’])
[20]: Jarque_beraResult(statistic=302.1619199254945, pvalue=0.0)
[21]: stats.jarque_bera(data_s[‘R_ca’])
[21]: Jarque_beraResult(statistic=2209.400046567293, pvalue=0.0)
[22]: stats.jarque_bera(data_s[‘R_c^2’])
[22]: Jarque_beraResult(statistic=197064.76373846942, pvalue=0.0)
Interpretation: The JB tests show there is a non-normal distribution (p-value <0.05), given the
Copper’s daily returns is negatively skewed and has an excess Kurtosis of 2.68.
4 b(iv) Histogram
[23]: _ = plt.hist(data['R_c'],bins=100)
_ = plt.xlabel('OBS')
_ = plt.ylabel('R_c')
plt.show()
plt.savefig('C:\\Users\\rluck\\OneDrive\\myplot.png')
3
[24]: _ = plt.hist(data[‘R_ca’],bins=100)
_ = plt.xlabel(‘OBS’)
_ = plt.ylabel(‘R_ca’)
plt.show()
4
[25]: _ = plt.hist(data[‘R_c^2’],bins=100)
_ = plt.xlabel(‘OBS’)
_ = plt.ylabel(‘R_c^2′)
plt.show()
5
5 Plotting charts
[26]: plt.plot(data.R_c,color=’red’)
[26]: [
[27]: plt.plot(data.R_ca,color=’blue’)
[27]: [
6
[28]: dta= data_s[‘R_c’]
6 Computing ACF and PACF
[29]: >>> sm.graphics.tsa.plot_pacf(dta.values.squeeze(), lags=24)
sm.graphics.tsa.plot_acf(dta.values.squeeze(),lags=24)
>>> plt.show()
7
8
[30]: r,q,p = sm.tsa.acf(dta.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=[‘lag’, “AC”, “Q”, “Prob(>Q)”])
print (table.set_index(‘lag’))
AC Q Prob(>Q)
lag
1.0 -0.169258 28.705576 8.426077e-08
2.0 0.013418 28.886150 5.338906e-07
3.0 -0.034927 30.110942 1.307854e-06
4.0 -0.042273 31.906892 1.998778e-06
5.0 0.040801 33.581630 2.884062e-06
6.0 -0.030759 34.534399 5.303518e-06
7.0 -0.048466 36.902332 4.894754e-06
8.0 0.008219 36.970498 1.165359e-05
9.0 0.023870 37.546022 2.102220e-05
10.0 -0.000546 37.546323 4.550870e-05
11.0 0.031456 38.547811 6.320925e-05
12.0 -0.019199 38.921279 1.084230e-04
13.0 -0.042012 40.711358 1.061207e-04
14.0 -0.010708 40.827757 1.893163e-04
15.0 0.050749 43.445094 1.342517e-04
16.0 0.012242 43.597563 2.270240e-04
17.0 -0.021301 44.059599 3.356420e-04
18.0 0.019068 44.430240 5.005839e-04
19.0 0.001336 44.432061 8.226156e-04
20.0 0.046795 46.668800 6.519565e-04
21.0 -0.017221 46.972047 9.472905e-04
22.0 0.060033 50.660881 4.770893e-04
23.0 -0.026672 51.389781 6.034195e-04
24.0 0.001243 51.391364 9.386307e-04
25.0 -0.062687 55.425856 4.326261e-04
26.0 -0.067126 60.056752 1.647766e-04
27.0 0.025273 60.713863 2.124971e-04
28.0 0.006739 60.760627 3.253423e-04
29.0 0.084367 68.098464 5.494580e-05
30.0 0.053573 71.060362 3.495973e-05
31.0 -0.068947 75.971216 1.206393e-05
32.0 0.011689 76.112502 1.858629e-05
33.0 -0.007772 76.175038 2.892027e-05
34.0 -0.074013 81.851574 7.997707e-06
35.0 0.068058 86.656475 2.858653e-06
36.0 -0.072387 92.097697 8.262371e-07
37.0 0.002966 92.106843 1.348620e-06
38.0 -0.004419 92.127165 2.164122e-06
39.0 0.003125 92.137336 3.438375e-06
40.0 0.014532 92.357529 5.063225e-06
9
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:657:
FutureWarning: The default number of lags is changing from 40 tomin(int(10 *
np.log10(nobs)), nobs – 1) after 0.12is released. Set the number of lags to an
integer to silence this warning.
warnings.warn(
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:667:
FutureWarning: fft=True will become the default after the release of the 0.12
release of statsmodels. To suppress this warning, explicitly set fft=False.
warnings.warn(
As per the above correlograms, the first autocorrelation of -0.169 is statistically significant as it
exceeds the bands (i.e: the blue shaded area) in the chart. We can infer that the log price of
Copper does not follow a random walk and the efficient market hypothesis does not hold. For
squared returns (R_c^2) and absolute returns (R_ca), the autocorrelations are even stronger, as
demonstrated by the following ACF and PACF charts.
[31]: dta_1= data_s[‘R_ca’]
[32]: >>> sm.graphics.tsa.plot_pacf(dta_1.values.squeeze(), lags=12)
sm.graphics.tsa.plot_acf(dta_1.values.squeeze(),lags=12)
>>> plt.show()
10
[33]: r,q,p = sm.tsa.acf(dta_1.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=[‘lag’, “AC”, “Q”, “Prob(>Q)”])
print (table.set_index(‘lag’))
AC Q Prob(>Q)
lag
1.0 0.250814 63.033756 2.031943e-15
2.0 0.152524 86.367442 1.760144e-19
3.0 0.153909 110.150402 1.018515e-23
4.0 0.127490 126.485836 2.196721e-26
5.0 0.150763 149.352358 1.833912e-30
6.0 0.140662 169.277473 6.400788e-34
7.0 0.109180 181.293948 1.038331e-35
8.0 0.098477 191.079671 4.827095e-37
9.0 0.100862 201.355470 1.722055e-38
10.0 0.119461 215.785175 8.144212e-41
11.0 0.068930 220.594251 3.883007e-41
12.0 0.056851 223.868911 3.739222e-41
13.0 0.059601 227.471661 2.983310e-41
14.0 0.078392 233.710486 6.632136e-42
15.0 0.076650 239.681269 1.646911e-42
16.0 0.074328 245.301502 4.753518e-43
17.0 0.079118 251.675918 9.486837e-44
18.0 0.109714 263.946256 1.175163e-45
11
19.0 0.103477 274.872190 2.735358e-47
20.0 0.085096 282.268874 3.322181e-48
21.0 0.129912 299.525625 3.972767e-51
22.0 0.105942 311.013572 7.090149e-53
23.0 0.107406 322.833230 1.082927e-54
24.0 0.096674 332.418789 4.696503e-56
25.0 0.108696 344.548874 6.198574e-58
26.0 0.096625 354.144379 2.668580e-59
27.0 0.110406 366.685037 2.907339e-61
28.0 0.090840 375.183395 2.081923e-62
29.0 0.088709 383.296077 1.779086e-63
30.0 0.088184 391.321299 1.578922e-64
31.0 0.133722 409.793997 1.092694e-67
32.0 0.066202 414.326274 4.907364e-68
33.0 0.129816 431.771517 5.489116e-71
34.0 0.089096 439.997554 4.430751e-72
35.0 0.102302 450.854107 1.054561e-73
36.0 0.064577 455.184466 5.153559e-74
37.0 0.076557 461.276941 1.108424e-74
38.0 0.080198 467.969607 1.800223e-75
39.0 0.108762 480.291745 2.170477e-77
40.0 0.049010 482.796366 2.423205e-77
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:657:
FutureWarning: The default number of lags is changing from 40 tomin(int(10 *
np.log10(nobs)), nobs – 1) after 0.12is released. Set the number of lags to an
integer to silence this warning.
warnings.warn(
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:667:
FutureWarning: fft=True will become the default after the release of the 0.12
release of statsmodels. To suppress this warning, explicitly set fft=False.
warnings.warn(
[34]: dta_2= data_s[‘R_c^2’]
[35]: >>> sm.graphics.tsa.plot_pacf(dta_2.values.squeeze(), lags=12)
sm.graphics.tsa.plot_acf(dta_2.values.squeeze(),lags=12)
>>> plt.show()
12
13
[36]: r,q,p = sm.tsa.acf(dta_2.values.squeeze(), qstat=True)
data = np.c_[range(1,41), r[1:], q, p]
table = pd.DataFrame(data, columns=[‘lag’, “AC”, “Q”, “Prob(>Q)”])
print (table.set_index(‘lag’))
AC Q Prob(>Q)
lag
1.0 0.267265 71.573667 2.670975e-17
2.0 0.136738 90.327290 2.430403e-20
3.0 0.111404 102.788031 3.907861e-22
4.0 0.099766 112.791242 1.847446e-23
5.0 0.120752 127.460370 8.229911e-26
6.0 0.125132 143.228817 2.086198e-28
7.0 0.066961 147.748703 1.205533e-28
8.0 0.055213 150.824826 1.319382e-28
9.0 0.100740 161.075827 4.438458e-30
10.0 0.091719 169.581739 3.386147e-31
11.0 0.032893 170.676845 8.560979e-31
12.0 0.031368 171.673737 2.169953e-30
13.0 0.039585 173.262956 4.004580e-30
14.0 0.051436 175.948865 4.288407e-30
15.0 0.038930 177.489023 7.623657e-30
16.0 0.009632 177.583403 2.568328e-29
17.0 0.057397 180.938148 1.876983e-29
18.0 0.096342 190.399625 8.250680e-31
19.0 0.081927 197.248691 1.199337e-31
20.0 0.058761 200.775589 7.903797e-32
21.0 0.110171 213.186262 9.044721e-34
22.0 0.082405 220.136703 1.246595e-34
23.0 0.077137 226.233201 2.526215e-35
24.0 0.065301 230.606764 1.113264e-35
25.0 0.034623 231.837496 2.011860e-35
26.0 0.044748 233.895403 2.468183e-35
27.0 0.090672 242.353562 1.698376e-36
28.0 0.102785 253.233828 3.941894e-38
29.0 0.057602 256.654487 2.599329e-38
30.0 0.017237 256.961123 6.832310e-38
31.0 0.073249 262.503968 1.722169e-38
32.0 0.011442 262.639345 4.777961e-38
33.0 0.063586 266.824825 2.179022e-38
34.0 0.048306 269.242934 2.159314e-38
35.0 0.044036 271.254528 2.539467e-38
36.0 0.049462 273.795001 2.349134e-38
37.0 0.043425 275.755218 2.782275e-38
38.0 0.027704 276.553889 5.423483e-38
39.0 0.041901 278.382719 6.688394e-38
40.0 0.020043 278.801625 1.506677e-37
14
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:657:
FutureWarning: The default number of lags is changing from 40 tomin(int(10 *
np.log10(nobs)), nobs – 1) after 0.12is released. Set the number of lags to an
integer to silence this warning.
warnings.warn(
C:\Users\rluck\anaconda3\lib\site-packages\statsmodels\tsa\stattools.py:667:
FutureWarning: fft=True will become the default after the release of the 0.12
release of statsmodels. To suppress this warning, explicitly set fft=False.
warnings.warn(
15
(a) Calculating the log returns of Copper
(b) (i) Summary statistics
b(iii) Jacques-Bera test
b(iv) Histogram
Plotting charts
Computing ACF and PACF