June 24, 2021
1 Calculate the daily log returns of T-Bill, gold, GE stock and
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[92]: import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
[93]: data = pd.read_csv(“C:\\Users\\rluck\\OneDrive\\capm.csv”, header=[4])
[93]: DATE Gold S&P500 Rf GE
0 12/08/1975 166.05 87.12 6.40 0.9218
1 13/08/1975 163.50 85.97 6.45 0.9036
2 14/08/1975 163.50 85.60 6.45 0.9036
3 15/08/1975 163.50 86.36 6.42 0.9244
4 18/08/1975 163.50 86.20 6.42 0.9348
… … … … … …
10432 6/08/2015 1090.15 2083.56 0.04 26.0300
10433 7/08/2015 1096.85 2077.57 0.06 25.7900
10434 10/08/2015 1103.35 2104.18 0.12 26.2400
10435 11/08/2015 1109.67 2084.07 0.10 25.7100
10436 12/08/2015 1123.85 2086.05 0.10 25.8600
[10437 rows x 5 columns]
#Computing log returns: R_gold = 100*ln(P_g/P_g-1)
Rf = 100/360*ln(1+rf)
[94]: data[‘R_gold’]=100*np.log(data[‘Gold’]/data[‘Gold’].shift(1))
data[‘R_f’]= 100/360*np.log(1+data[‘Rf’]/100)
data[‘R_GE’]= 100*np.log(data[‘GE’]/data[‘GE’].shift(1))
data[‘R_m’]= 100*np.log(data[‘S&P500’]/data[‘S&P500’].shift(1))
print(data.head())
DATE Gold S&P500 Rf GE R_gold R_f R_GE \
0 12/08/1975 166.05 87.12 6.40 0.9218 NaN 0.017232 NaN
1 13/08/1975 163.50 85.97 6.45 0.9036 -1.547596 0.017363 -1.994150
2 14/08/1975 163.50 85.60 6.45 0.9036 0.000000 0.017363 0.000000
3 15/08/1975 163.50 86.36 6.42 0.9244 0.000000 0.017284 2.275809
4 18/08/1975 163.50 86.20 6.42 0.9348 0.000000 0.017284 1.118772
1 -1.328808
2 -0.431312
3 0.883932
4 -0.185443
2 Calculating excess returns for gold and GE
[95]: data[‘R_p’]= data[‘R_m’]- data[‘R_f’]
data[‘R_ge’]= data[‘R_GE’]-data[‘R_f’]
data[‘R_go’]= data[‘R_gold’]-data[‘R_f’]
[95]: DATE Gold S&P500 Rf GE R_gold R_f \
0 12/08/1975 166.05 87.12 6.40 0.9218 NaN 0.017232
1 13/08/1975 163.50 85.97 6.45 0.9036 -1.547596 0.017363
2 14/08/1975 163.50 85.60 6.45 0.9036 0.000000 0.017363
3 15/08/1975 163.50 86.36 6.42 0.9244 0.000000 0.017284
4 18/08/1975 163.50 86.20 6.42 0.9348 0.000000 0.017284
… … … … … … … …
10432 6/08/2015 1090.15 2083.56 0.04 26.0300 0.415484 0.000111
10433 7/08/2015 1096.85 2077.57 0.06 25.7900 0.612713 0.000167
10434 10/08/2015 1103.35 2104.18 0.12 26.2400 0.590857 0.000333
10435 11/08/2015 1109.67 2084.07 0.10 25.7100 0.571167 0.000278
10436 12/08/2015 1123.85 2086.05 0.10 25.8600 1.269762 0.000278
R_GE R_m R_p R_ge R_go
0 NaN NaN NaN NaN NaN
1 -1.994150 -1.328808 -1.346171 -2.011512 -1.564958
2 0.000000 -0.431312 -0.448674 -0.017363 -0.017363
3 2.275809 0.883932 0.866648 2.258525 -0.017284
4 1.118772 -0.185443 -0.202727 1.101488 -0.017284
… … … … … …
10432 -0.268560 -0.778318 -0.778429 -0.268671 0.415373
10433 -0.926290 -0.287903 -0.288069 -0.926457 0.612547
10434 1.729814 1.272690 1.272357 1.729481 0.590524
10435 -2.040494 -0.960313 -0.960591 -2.040772 0.570889
10436 0.581735 0.094961 0.094684 0.581458 1.269484
[10437 rows x 12 columns]
3 Data : Remove N/A
[96]: data = data.dropna(subset=[“R_p”])
data.to_csv(“C:\\Users\\rluck\\OneDrive\\capm1.csv”)
data.head()
[96]: DATE Gold S&P500 Rf GE R_gold R_f R_GE \
1 13/08/1975 163.5 85.97 6.45 0.9036 -1.547596 0.017363 -1.994150
2 14/08/1975 163.5 85.60 6.45 0.9036 0.000000 0.017363 0.000000
3 15/08/1975 163.5 86.36 6.42 0.9244 0.000000 0.017284 2.275809
4 18/08/1975 163.5 86.20 6.42 0.9348 0.000000 0.017284 1.118772
5 19/08/1975 163.5 84.95 6.47 0.9218 0.000000 0.017415 -1.400432
R_m R_p R_ge R_go
1 -1.328808 -1.346171 -2.011512 -1.564958
2 -0.431312 -0.448674 -0.017363 -0.017363
3 0.883932 0.866648 2.258525 -0.017284
4 -0.185443 -0.202727 1.101488 -0.017284
5 -1.460733 -1.478148 -1.417847 -0.017415
[97]: !pip install sklearn
!pip install statsmodels
Requirement already satisfied: sklearn in c:\users\rluck\anaconda3\lib\site-
packages (0.0)
Requirement already satisfied: scikit-learn in
c:\users\rluck\anaconda3\lib\site-packages (from sklearn) (0.24.1)
Requirement already satisfied: scipy>=0.19.1 in
c:\users\rluck\anaconda3\lib\site-packages (from scikit-learn->sklearn) (1.6.2)
Requirement already satisfied: joblib>=0.11 in
c:\users\rluck\anaconda3\lib\site-packages (from scikit-learn->sklearn) (1.0.1)
Requirement already satisfied: numpy>=1.13.3 in
c:\users\rluck\anaconda3\lib\site-packages (from scikit-learn->sklearn) (1.20.1)
Requirement already satisfied: threadpoolctl>=2.0.0 in
c:\users\rluck\anaconda3\lib\site-packages (from scikit-learn->sklearn) (2.1.0)
Requirement already satisfied: statsmodels in c:\users\rluck\anaconda3\lib\site-
packages (0.12.2)
Requirement already satisfied: numpy>=1.15 in c:\users\rluck\anaconda3\lib\site-
packages (from statsmodels) (1.20.1)
Requirement already satisfied: scipy>=1.1 in c:\users\rluck\anaconda3\lib\site-
packages (from statsmodels) (1.6.2)
Requirement already satisfied: pandas>=0.21 in
c:\users\rluck\anaconda3\lib\site-packages (from statsmodels) (1.2.4)
Requirement already satisfied: patsy>=0.5 in c:\users\rluck\anaconda3\lib\site-
packages (from statsmodels) (0.5.1)
Requirement already satisfied: python-dateutil>=2.7.3 in
c:\users\rluck\anaconda3\lib\site-packages (from pandas>=0.21->statsmodels)
Requirement already satisfied: pytz>=2017.3 in
c:\users\rluck\anaconda3\lib\site-packages (from pandas>=0.21->statsmodels)
Requirement already satisfied: six in c:\users\rluck\anaconda3\lib\site-packages
(from patsy>=0.5->statsmodels) (1.15.0)
[98]: %matplotlib inline
import statsmodels.api as sm
import statsmodels.formula.api as smf
from sklearn import linear_model
import matplotlib.pyplot as plt
4 I. Plotting Gold excess returns with market excess returns
[99]: #Regressing excess returns on gold (R_g-Rf) over risk-free rate against the␣
↪→excess market return (Rp=Rm-rf)
reg = linear_model.LinearRegression()
X =data[[‘R_p’]].dropna()
y1 =data[‘R_go’].dropna()
reg.fit(X,y1)
predictions =reg.predict(X)
[100]: plt.xlabel(‘Risk Premium’)
plt.ylabel(‘R_go’)
plt.scatter(data.R_p,data.R_gold,color=’red’,marker=’+’)
plt.plot(data.R_p,reg.predict(data[[‘R_p’]]), color=’orange’)
[100]: [
[101]: #model with intercept
X= sm.add_constant(X)
model = sm.OLS(y1,X).fit()
predictions = model.predict(X)
j= (model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: R_go R-squared: 0.000
Model: OLS Adj. R-squared: 0.000
Method: Least Squares F-statistic: 3.181
Date: Thu, 24 Jun 2021 Prob (F-statistic): 0.0745
Time: 13:42:13 Log-Likelihood: -16959.
No. Observations: 10436 AIC: 3.392e+04
Df Residuals: 10434 BIC: 3.394e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
——————————————————————————
const 0.0057 0.012 0.473 0.637 -0.018 0.029
R_p -0.0201 0.011 -1.784 0.075 -0.042 0.002
==============================================================================
Omnibus: 2812.110 Durbin-Watson: 2.071
Prob(Omnibus): 0.000 Jarque-Bera (JB): 111926.422
Skew: -0.573 Prob(JB): 0.00
Kurtosis: 19.003 Cond. No. 1.07
==============================================================================
[1] Standard Errors assume that the covariance matrix of the errors is correctly
specified.
DW-stats of 2.071 is close to 2.0, implying that there is no serial correlation.
Yet, the p-value of the beta coefficient indicates that it is slightly significant at 7.5% significance
level and the R-squared is very low, explaining low explanatory power of the model.
5 Residuals plot for gold
[102]: residuals_go = model.resid
import seaborn as sns
sns.distplot(residuals_go,hist=True, kde=True, bins=int(120), color=␣
↪→’darkblue’,hist_kws={‘edgecolor’:’black’})
C:\Users\rluck\anaconda3\lib\site-packages\seaborn\distributions.py:2557:
FutureWarning: `distplot` is a deprecated function and will be removed in a
future version. Please adapt your code to use either `displot` (a figure-level
function with similar flexibility) or `histplot` (an axes-level function for
histograms).
warnings.warn(msg, FutureWarning)
[102]:
[103]: from scipy import stats
JB_go= stats.jarque_bera(residuals_go)
[103]: Jarque_beraResult(statistic=111926.42195044507, pvalue=0.0)
The plot and JB test (p-value <0.05) rejects the null hypothesis of normality. It is clearly a
non-normal distribution.
6 Cusum Test for Gold
[104]: # endog = data.R_go
Rp = data.R_p
endog = data.R_go
exog = sm.add_constant(Rp)
mod = sm.RecursiveLS(endog,exog)
res_1 = mod.fit()
fig = res_1.plot_cusum(figsize=(10,6));
C:\Users\rluck\anaconda3\lib\site-
packages\statsmodels\tsa\base\tsa_model.py:578: ValueWarning: An unsupported
index was provided and will be ignored when e.g. forecasting.
warnings.warn('An unsupported index was provided and will be'
Cusum test of stability for gold shows high periods of instability during the early part of the graph
(namely before 1980s).Then, the beta stabilises.
7 White Test of Heteroskedasticity for Gold
[105]: from statsmodels.stats.diagnostic import het_white
from statsmodels.compat import lzip
from patsy import dmatrices
[106]: expr = 'y1 ~ X'
y1, X = dmatrices(expr, data, return_type='dataframe')
olsr_results = smf.ols(expr, data).fit()
keys = [' statistic:', 'LM test\'s p-value:', 'F-statistic:
↪→', 'F-test\'s p-value:']
results = het_white(olsr_results.resid, X)
lzip(keys, results)
[106]: [(' statistic:', 36.866651106570274),
("LM test's p-value:", 9.874348003656595e-09),
('F-statistic:', 18.49335699862814),
("F-test's p-value:", 9.608158442586967e-09)]
LM test statistic is 36.87 and the corresponding p-value is 0
F-stats = 18.49 and the corresponding p-value is 0
Since the p-value of the both LM and F-stats is less than 0.05, we reject the null hypothesis that
there is no heteroskedasticity in the residuals. It infers that the heteroskedasticity exists and the
standard errors need to be corrected.
8 Breusch- M test for Gold
[107]: import statsmodels.stats.diagnostic as dg
print (dg.acorr_breusch_godfrey(model, nlags= 2))
(14.058774886495657, 0.0008854740175917412, 7.036171882380294,
0.0008836668869260258)
T-statistic of Chi-squared is 14.0588 and the corresponding p-value is 0.0009
F-statistic is 7.0362 and the corresponding p-value is 0.0009
Since p-value is less than 0.05, we reject the null hypothesis, thus inferring there is some autocor-
relation at order less than or equal to 2.0
9 II. Plotting GE excess returns with market excess returns
[108]: %matplotlib inline
reg = linear_model.LinearRegression()
X =data[['R_p']]
y =data['R_ge']
reg.fit(X,y)
[108]: LinearRegression()
[109]: plt.xlabel('Risk Premium')
plt.ylabel('R_GE')
plt.scatter(data.R_p,data.R_GE,color='green',marker='+')
plt.plot(data.R_p,reg.predict(data[['R_p']]), color='red')
[109]: [
10 Regressing GE excess return with market excess return
[110]: #model with intercept
X =sm.add_constant(X)
model_1 = sm.OLS(y,X).fit()
predictions = model_1.predict(X)
j= (model_1.summary())
OLS Regression Results
==============================================================================
Dep. Variable: R_ge R-squared: 0.564
Model: OLS Adj. R-squared: 0.564
Method: Least Squares F-statistic: 1.351e+04
Date: Thu, 24 Jun 2021 Prob (F-statistic): 0.00
Time: 13:42:14 Log-Likelihood: -15682.
No. Observations: 10436 AIC: 3.137e+04
Df Residuals: 10434 BIC: 3.138e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
——————————————————————————
const -0.0012 0.011 -0.114 0.909 -0.022 0.020
R_p 1.1569 0.010 116.224 0.000 1.137 1.176
==============================================================================
Omnibus: 2325.123 Durbin-Watson: 1.995
Prob(Omnibus): 0.000 Jarque-Bera (JB): 109234.319
Skew: 0.109 Prob(JB): 0.00
Kurtosis: 18.848 Cond. No. 1.07
==============================================================================
[1] Standard Errors assume that the covariance matrix of the errors is correctly
specified.
DW-stats of 1.995 is close to 2.0, implying that there is no serial correlation.
Since p-value of the beta coefficient is less than 0.05, we reject the null hypothesis that beta is zero.
The CAPM equation for GE can be written as follows:
Rge = 1.1569 ∗Rp +Rf
where Rge is the return from GE stock, Rp = Rm−Rf is the market risk premium and Rf is the
risk free rate of return
If we want to replicate the returns from GE, we can rearrange the above equation:
Rge = 1.1569 ∗Rm+ (1− 1.1569) ∗Rf
⇒ We can buy 1.1569 of market portfolio (i.e: S&P500 index fund) and then short 0.1569 T-Bill.
11 Residual Plots for GE
[111]: residuals = model_1.resid
import seaborn as sns
sns.distplot(residuals,hist=True, kde=True, bins=int(120), color=␣
↪→’darkblue’,hist_kws={‘edgecolor’:’black’})
C:\Users\rluck\anaconda3\lib\site-packages\seaborn\distributions.py:2557:
FutureWarning: `distplot` is a deprecated function and will be removed in a
future version. Please adapt your code to use either `displot` (a figure-level
function with similar flexibility) or `histplot` (an axes-level function for
histograms).
warnings.warn(msg, FutureWarning)
[111]:
[112]: from scipy import stats
JB_GE= stats.jarque_bera(residuals)
[112]: Jarque_beraResult(statistic=109234.31887176927, pvalue=0.0)
The plot and JB test (p-value <0.05) rejects the null hypothesis of normality. It is clearly a non-normal distribution. [113]: endog = data.R_ge Rp = data.R_p exog = sm.add_constant(Rp) mod = sm.RecursiveLS(endog,exog) res_1 = mod.fit() fig = res_1.plot_cusum(figsize=(10,6)); C:\Users\rluck\anaconda3\lib\site- packages\statsmodels\tsa\base\tsa_model.py:578: ValueWarning: An unsupported index was provided and will be ignored when e.g. forecasting. warnings.warn('An unsupported index was provided and will be' Cusum test of stability for GE shows stability of beta as it is within the 5% significance level band. 12 White Test of Heteroskedasticity for GE [114]: expr = 'y ~ X' y, X = dmatrices(expr, data, return_type='dataframe') olsr_results = smf.ols(expr, data).fit() keys = [' statistic:', 'LM test\'s p-value:', 'F-statistic: ↪→', 'F-test\'s p-value:'] results = het_white(olsr_results.resid, X) lzip(keys, results) [114]: [(' statistic:', 600.7119211665138), ("LM test's p-value:", 3.606315445696676e-131), ('F-statistic:', 318.60924780728743), ("F-test's p-value:", 4.9073934718673876e-135)] LM test statistic is 600.72 and the corresponding p-value is 0 F-stats = 318.61 and the corresponding p-value is 0 Since the p-value of the both LM and F-stats is less than 0.05, we reject the null hypothesis that there is no heteroskedasticity in the residuals. It infers that the heteroskedasticity exists and the standard errors need to be corrected. [115]: print (dg.acorr_breusch_godfrey(model_1, nlags= 2)) (5.174836714176367, 0.07521396525512013, 2.587709781212114, 0.07524031416320724) T-statistic of Chi-squared = 5.1748 and the corresponding p-value = 0.075. F-statistics = 2.5877 and the corresponding p-value = 0.075 Since p-value exceeds 0.05, we fail to reject the null hypothesis, thus inferring there is no autocor- relation at order less than or equal to 2.0 13 Extra: Correlation matrix between returns of gold, GE and [116]: import seaborn as sns import pandas as pd # Import Dataset data = pd.read_csv("C:\\Users\\rluck\\OneDrive\\capm1.csv",␣ ↪→usecols=['R_gold','R_m', 'R_GE']) plt.figure(figsize=(12,10), dpi= 80) sns.heatmap(data.corr(), xticklabels=data.corr().columns, yticklabels=data. ↪→corr().columns, cmap='RdYlGn', center=0, annot=True) # Decorations plt.title('Correlogram of returns', fontsize=22) plt.xticks(fontsize=12) plt.yticks(fontsize=12) plt.show() Calculate the daily log returns of T-Bill, gold, GE stock and market Calculating excess returns for gold and GE Data : Remove N/A I. Plotting Gold excess returns with market excess returns Residuals plot for gold Cusum Test for Gold White Test of Heteroskedasticity for Gold Breusch- M test for Gold II. Plotting GE excess returns with market excess returns Regressing GE excess return with market excess return Residual Plots for GE White Test of Heteroskedasticity for GE Extra: Correlation matrix between returns of gold, GE and market 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com