Exercises: Integration, Cointegration, and Stationarity - Answer Key

by Delaney Granizo-Mackenzie and Maxwell Margenot

Lecture Link :

https://www.quantopian.com/lectures/integration-cointegration-and-stationarity

IMPORTANT NOTE:

This lecture corresponds to the Integration, Cointegration, and Stationarity lecture, which is part of the Quantopian lecture series. This homework expects you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult.

Part of the Quantopian Lecture Series:

Helper Functions



In [1]:

    
# Useful Functions
def check_for_stationarity(X, cutoff=0.01):
    # H_0 in adfuller is unit root exists (non-stationary)
    # We must observe significant p-value to convince ourselves that the series is stationary
    pvalue = adfuller(X)[1]
    if pvalue < cutoff:
        print 'p-value = ' + str(pvalue) + ' The series is likely stationary.'
        return True
    else:
        print 'p-value = ' + str(pvalue) + ' The series is likely non-stationary.'
        return False
    
def generate_datapoint(params):
    mu = params[0]
    sigma = params[1]
    return np.random.normal(mu, sigma)



In [2]:

    
# Useful Libraries
import numpy as np
import pandas as pd

import statsmodels
import statsmodels.api as sm
from statsmodels.tsa.stattools import coint, adfuller

import matplotlib.pyplot as plt

Exercise 1: Stationarity Testing

a. Checking For Stationarity

Check whether the following series is stationary using the tests from the lecture.



In [3]:

    
QQQ = get_pricing("QQQ", start_date='2014-1-1', end_date='2015-1-1', fields='price')
QQQ.name = QQQ.name.symbol

check_for_stationarity(QQQ)









    



p-value = 0.83074729803 The series is likely non-stationary.






    Out[3]:





False

b. Checking for Normality

As an extra all-purpose check, and one that is often done on series, check whether the above series is normally distributed using the Jarque-Bera test.



In [4]:

    
from statsmodels.stats.stattools import jarque_bera

jarque_bera(QQQ)









    Out[4]:





(16.77608340090589,
 0.00022757249739878912,
 0.2415274043817217,
 1.8319331010647477)

c. Constructing Examples I

Create/provide a series that is stationary and different from any covered so far in the exercise or the lecture.



In [5]:

    
X = np.random.normal(0, 1, 100)

check_for_stationarity(X)









    



p-value = 8.15924515037e-16 The series is likely stationary.






    Out[5]:





True

d. Constructing Examples II

Create/provide a series that is non-stationary and different from any covered so far in the exercise or the lecture.



In [6]:

    
# Set the number of datapoints
T = 100

B = pd.Series(index=range(T))
B.name = 'B'

for t in range(T):
    # Now the parameters are dependent on time
    # Specifically, the mean of the series changes over time
    params = (np.power(t, 2), 1)
    B[t] = generate_datapoint(params)



In [7]:

    
plt.plot(B);



In [8]:

    
check_for_stationarity(B)









    



p-value = 0.964313099004 The series is likely non-stationary.






    Out[8]:





False

Exercise 2: Estimate Order of Integration

Use the techniques laid out in the lecture notebook to estimate the order of integration for the following timeseries.



In [9]:

    
QQQ = get_pricing("QQQ", start_date='2014-1-1', end_date='2015-1-1', fields='price')
QQQ.name = QQQ.name.symbol

# Write code to estimate the order of integration of QQQ.
# Feel free to sample from the code provided in the lecture.



In [10]:

    
QQQ = QQQ.diff()[1:]
QQQ.name = QQQ.name + ' Additive Returns'
check_for_stationarity(QQQ)
plt.plot(QQQ.index, QQQ.values)
plt.ylabel('Additive Returns')
plt.legend([QQQ.name]);









    



p-value = 2.93736515664e-28 The series is likely stationary.

Exercise 3: Find a Stationary Linear (Cointegrated) Combination

Use the techniques laid out in the lecture notebook to find a linear combination of the following timeseries that is stationary.



In [11]:

    
T = 500

X1 = pd.Series(index=range(T))
X1.name = 'X1'

for t in range(T):
    # Now the parameters are dependent on time
    # Specifically, the mean of the series changes over time
    params = (t * 0.1, 1)
    X1[t] = generate_datapoint(params)

X2 = np.power(X1, 2) + X1
X3 = np.power(X1, 3) + X1
X4 = np.sin(X1) + X1

# We now have 4 time series, X1, X2, X3, X4
# Determine a linear combination of the 4 that is stationary over the 
# time period shown using the techniques in the lecture.



In [12]:

    
X1 = sm.add_constant(X1)
results = sm.OLS(X4, X1).fit()

# Get rid of the constant column
X1 = X1['X1']

results.params









    Out[12]:





const    0.017736
X1       0.997973
dtype: float64



In [13]:

    
plt.plot(X4-0.99 * X1);



In [14]:

    
check_for_stationarity(X4 - 0.99*X1)









    



p-value = 7.40184822771e-07 The series is likely stationary.






    Out[14]:





True

Congratulations on completing the Integration, Cointegration, and Stationarity answer key!

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