Calculation of the Half-Life of a Mean-Reverting Time Series

In [1]:

    

import numpy as np


    

In [2]:

    

import pandas as pd


    

In [3]:

    

import matplotlib.pyplot as plt


    

In [4]:

    

from statsmodels.tsa.stattools import coint


    

In [5]:

    

from statsmodels.api import OLS


    

In [6]:

    

df1=pd.read_excel('GLD.xls')


    

In [7]:

    

df2=pd.read_excel('GDX.xls')


    

In [8]:

    

df=pd.merge(df1, df2, on='Date', suffixes=('_GLD', '_GDX'))


    

In [9]:

    

df.set_index('Date', inplace=True)


    

In [10]:

    

df.sort_index(inplace=True)


    

In [11]:

    

coint_t, pvalue, crit_value=coint(df['Adj Close_GLD'], df['Adj Close_GDX'])


    

In [12]:

    

(coint_t, pvalue, crit_value) # abs(t-stat) > critical value at 95%. pvalue says probability of null hypothesis (of no cointegration) is only 1.8%


    

    
Out[12]:





(-3.6981160763300593,
 0.018427835409537425,
 array([-3.92518794, -3.35208799, -3.05551324]))

In [13]:

    

model=OLS(df['Adj Close_GLD'], df['Adj Close_GDX'])


    

In [14]:

    

results=model.fit()


    

In [15]:

    

hedgeRatio=results.params


    

In [16]:

    

hedgeRatio


    

    
Out[16]:





Adj Close_GDX    1.639523
dtype: float64

In [17]:

    

z=df['Adj Close_GLD']-hedgeRatio[0]*df['Adj Close_GDX']


    

In [18]:

    

plt.plot(z)


    

    
Out[18]:





[<matplotlib.lines.Line2D at 0x22d02eceb00>]






    

In [20]:

    

prevz=z.shift()


    

In [23]:

    

dz=z-prevz


    

In [27]:

    

dz=dz[1:,]


    

In [29]:

    

prevz=prevz[1:,]


    

In [38]:

    

model2=OLS(dz, prevz-np.mean(prevz))


    

In [39]:

    

results2=model2.fit()


    

In [40]:

    

theta=results2.params


    

In [41]:

    

theta


    

    
Out[41]:





x1   -0.088423
dtype: float64

In [43]:

    

halflife=-np.log(2)/theta


    

In [44]:

    

halflife


    

    
Out[44]:





x1    7.839031
dtype: float64