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How To Pick A Good Cointegrating Pair

Introduction

A time series is considered stationary if its probability distribution does not change over time. If the price series of a security is stationary, then it would be a suitable candidate for a mean-reversion trading strategy. However, most security price series are not stationary: they seem to follow a lognormal random walk; and drift farther and farther away from the initial value.

We need to find a pair of securities such that the combination of the two is stationary, e.g. buying a security and shorting another. Two securities that form a stationary or cointegrating pair are often from the same industry group such as Coca-Cola Company and PepsiCo. In this article, we illustrate how to pick a good cointegrating pair by applying the augmented Dickey-Fuller test to security pairs to check for cointegration.

Step-by-step

We will proceed as follows:

  1. Determine The Pairs: We present the security pairs to analyze.
  2. Prepare The Data: We pull and process securities' open-high-low-close-volume (OHLCV) data.
  3. Calculate The Spread: We apply the ordinary least squares (OLS) method to calculate the spread between two securities.
  4. Check For Cointegration: We use the augmented Dickey-Fuller test to check if two securities form a stationary or cointegrating pair.

You can find the code on https://github.com/DinodC/cointegrating-pair.

Determine The Pairs

Below are the pairs of securities which we will check for cointegration:

1. Gold

Gold-themed exchange traded funds (ETF):

  • VanEck Vectors Gold Miners ETF (GDX): ETF which tracks a basket of gold-mining companies.
  • SPDR Gold Shares (GLD): ETF which replicates the price of gold bullion.

2. Fast Food

Companies serving fast food:

  • McDonald's Corporation (MCD): Fast food company which gave the whole world classics like Big Mac, Hot Fudge Sundae, and Happy Meal.
  • YUM! Brands, Inc. (YUM): Fast food company which operates Taco Bell, KFC and Pizza Hut.

3. Cryptocurrencies

Digital currencies:

  • Bitcoin USD (BTC-USD): A decentralized cryptocurrency that can be sent from user to user on the peer-to-peer bitcoin network established in 2009.
  • Ethereum USD (ETH-USD): An open source, public, blockchain-based distributed computing platform and operating system released in 2015.

Prepare The Data

In this section, we illustrate download and preparation of securities' price series. We pull the securities' historical OHLCV data from Yahoo Finance. We select the adjusted close prices for each security and create a new Dataframe object.

Import packages

import pandas as pd
from pandas import DataFrame
from statsmodels.tsa.stattools import adfuller
import statsmodels.api as sm
import matplotlib.pyplot as plt

Magic

%matplotlib inline

Set tickers list

tickers = ['GDX', 'GLD', 'MCD', 'YUM', 'BTC-USD', 'ETH-USD']

Pull OHLCV data

# Initialize list of DataFrames
df_list = []

# Load DataFrames
for i in tickers:

    # Load data
    df = pd.read_csv(i + '.csv', index_col=0, parse_dates=True)    

    # Set multi-level columns
    df.columns = pd.MultiIndex.from_product([[i], ['Open', 'High', 'Low', 'Close', 'Adj Close', 'Volume']])

    # Update list
    df_list.append(df)

# Merge DataFrames
data = pd.concat(df_list, axis=1, join='inner')

# Drop NaNs
data.dropna(inplace=True)

Inspect OHLCV data

data.head()
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GDX GLD ... BTC-USD ETH-USD
Open High Low Close Adj Close Volume Open High Low Close ... Low Close Adj Close Volume Open High Low Close Adj Close Volume
Date
2015-08-06 13.21 13.69 13.11 13.36 13.033523 69121200 104.150002 104.860001 104.139999 104.389999 ... 274.279999 277.890015 277.890015 11919665 0.6747 3.000 0.6747 3.000 3.000 371
2015-08-07 13.42 13.85 13.33 13.40 13.072546 50618200 104.559998 105.379997 104.550003 104.650002 ... 257.420013 258.600006 258.600006 22308123 3.0000 3.000 0.1500 1.200 1.200 1438
2015-08-10 13.57 14.29 13.36 14.27 13.921287 91376800 105.029999 106.269997 104.919998 105.720001 ... 261.440002 269.029999 269.029999 13681939 1.2000 1.200 0.6504 0.990 0.990 7419
2015-08-11 14.44 14.53 13.94 14.53 14.174931 53731900 106.489998 106.629997 105.769997 106.260002 ... 263.660004 267.660004 267.660004 15232934 0.9900 1.288 0.9050 1.288 1.288 2376
2015-08-12 14.81 15.53 14.78 15.52 15.140740 123217200 106.989998 107.910004 106.930000 107.750000 ... 261.279999 263.440002 263.440002 14962211 1.2880 1.885 1.2630 1.885 1.885 4923

5 rows × 36 columns

For WordPress

data.head()[['GDX']]
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GDX
Open High Low Close Adj Close Volume
Date
2015-08-06 13.21 13.69 13.11 13.36 13.033523 69121200
2015-08-07 13.42 13.85 13.33 13.40 13.072546 50618200
2015-08-10 13.57 14.29 13.36 14.27 13.921287 91376800
2015-08-11 14.44 14.53 13.94 14.53 14.174931 53731900
2015-08-12 14.81 15.53 14.78 15.52 15.140740 123217200 
data.tail()
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GDX GLD ... BTC-USD ETH-USD
Open High Low Close Adj Close Volume Open High Low Close ... Low Close Adj Close Volume Open High Low Close Adj Close Volume
Date
2019-07-08 25.450001 25.610001 25.209999 25.420000 25.420000 40606100 132.179993 132.339996 131.279999 131.289993 ... 12117.309570 12567.019531 12567.019531 993891866 313.339996 318.320007 303.089996 307.890015 307.890015 125850428
2019-07-09 25.330000 25.660000 25.209999 25.650000 25.650000 37529700 131.429993 132.100006 131.160004 131.750000 ... 11569.940430 12099.120117 12099.120117 1554955347 307.890015 314.739990 281.619995 288.640015 288.640015 180940011
2019-07-10 26.020000 26.230000 25.770000 26.200001 26.200001 56454300 132.940002 133.869995 132.350006 133.830002 ... 11002.389648 11343.120117 11343.120117 1185222449 288.640015 288.660004 263.000000 268.559998 268.559998 171079615
2019-07-11 26.129999 26.280001 25.719999 25.940001 25.940001 54013400 133.580002 133.699997 132.410004 132.699997 ... 11096.610352 11797.370117 11797.370117 647690095 268.559998 279.059998 266.459991 275.410004 275.410004 76685542
2019-07-12 26.000000 26.250000 25.870001 26.209999 26.209999 31795200 132.889999 133.690002 132.529999 133.529999 ... 10827.530273 11363.969727 11363.969727 668325183 275.410004 275.720001 261.809998 268.940002 268.940002 66861426

5 rows × 36 columns

For WordPress

data.tail()[['GDX']]
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GDX
Open High Low Close Adj Close Volume
Date
2019-07-08 25.450001 25.610001 25.209999 25.420000 25.420000 40606100
2019-07-09 25.330000 25.660000 25.209999 25.650000 25.650000 37529700
2019-07-10 26.020000 26.230000 25.770000 26.200001 26.200001 56454300
2019-07-11 26.129999 26.280001 25.719999 25.940001 25.940001 54013400
2019-07-12 26.000000 26.250000 25.870001 26.209999 26.209999 31795200

Select adjusted close prices

# Initialize dictionary of adjusted close
close_dict = {}

# Update dictionary
for i in tickers:
    close_dict[i] = data[i]['Adj Close']

# Create DataFrame
close = pd.DataFrame(close_dict)

Inspect adjusted close prices

close.head()
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GDX GLD MCD YUM BTC-USD ETH-USD
Date
2015-08-06 13.033523 104.389999 89.038742 57.964733 277.890015 3.000
2015-08-07 13.072546 104.650002 88.653343 57.859062 258.600006 1.200
2015-08-10 13.921287 105.720001 89.074577 57.997757 269.029999 0.990
2015-08-11 14.174931 106.260002 88.554787 55.171165 267.660004 1.288
2015-08-12 15.140740 107.750000 88.079796 53.282372 263.440002 1.885 
close.tail()
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GDX GLD MCD YUM BTC-USD ETH-USD
Date
2019-07-08 25.420000 131.289993 212.160004 110.050003 12567.019531 307.890015
2019-07-09 25.650000 131.750000 212.089996 110.489998 12099.120117 288.640015
2019-07-10 26.200001 133.830002 213.000000 110.980003 11343.120117 268.559998
2019-07-11 25.940001 132.699997 212.690002 111.500000 11797.370117 275.410004
2019-07-12 26.209999 133.529999 212.990005 111.050003 11363.969727 268.940002

Consider the training set from 2018 to present

training = close['2018-01-01':'2020-01-01'].copy()

Inspect training set

training.head()
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GDX GLD MCD YUM BTC-USD ETH-USD
Date
2018-01-02 23.694632 125.150002 166.895370 79.503891 14754.129883 861.969971
2018-01-03 23.445948 124.820000 166.192001 79.435699 15156.620117 941.099976
2018-01-04 23.595158 125.459999 167.357834 80.244370 15180.080078 944.830017
2018-01-05 23.545422 125.330002 167.695084 80.712036 16954.779297 967.130005
2018-01-08 23.296738 125.309998 167.579422 80.848442 14976.169922 1136.109985 
training.tail()
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GDX GLD MCD YUM BTC-USD ETH-USD
Date
2019-07-08 25.420000 131.289993 212.160004 110.050003 12567.019531 307.890015
2019-07-09 25.650000 131.750000 212.089996 110.489998 12099.120117 288.640015
2019-07-10 26.200001 133.830002 213.000000 110.980003 11343.120117 268.559998
2019-07-11 25.940001 132.699997 212.690002 111.500000 11797.370117 275.410004
2019-07-12 26.209999 133.529999 212.990005 111.050003 11363.969727 268.940002

Calculate the number of pairs

no_pairs = round(0.5 * len(tickers))

Plot the adjusted close prices

plt.figure(figsize=(20, 20))

for i in range(no_pairs):
    # Primary axis
    color = 'tab:blue'
    ax1 = plt.subplot(3, 1, i+1)
    plt.plot(training[tickers[2*i]], color=color)
    ax1.set_ylabel('Adjusted Close Price of ' + tickers[2*i], color=color)
    ax1.tick_params(labelcolor=color)

    # Secondary axis
    color = 'tab:orange'
    ax2 = ax1.twinx()
    plt.plot(training[tickers[2*i+1]], color=color)
    ax2.set_ylabel('Adjusted Close Price of ' + tickers[2*i+1], color=color)
    ax2.tick_params(labelcolor=color)

    # Both axis
    plt.xlim([training.index[0], training.index[-1]])
    plt.title('Adjusted Close Prices of ' + tickers[2*i] + ' and ' + tickers[2*i+1])

png

Calculate The Spread

In this section, we calculate the spread between the securities. We apply the OLS method between the securities to calculate for the hedge ratio. We standardize the spread by subtracting the mean and scaling by the standard deviation of the spread.

Calculate the spread between each pair

# Initialize the spread list
spread_list = []

for i in range(no_pairs):
    # Run an OLS regression between the pairs
    model = sm.regression.linear_model.OLS(training[tickers[2*i]], training[tickers[2*i+1]])

    # Calculate the hedge ratio
    results = model.fit()
    hedge_ratio = results.params[0]

    # Calculate the spread
    spread = training[tickers[2*i]] - hedge_ratio * training[tickers[2*i+1]]

    # Mean and standard deviation of the spread
    spread_mean = spread.mean()
    spread_std = spread.std()

    # Standardize the spread
    z_score = (spread - spread_mean) / spread_std

    # Update the spread list
    spread_list.append(z_score)

Plot the spread

plt.figure(figsize=(20, 20))

for i in range(no_pairs):
    plt.subplot(3, 1, i+1)
    plt.plot(spread_list[i])
    plt.xlim([spread.index[0], spread.index[-1]])
    plt.ylim([-3, 3])
    plt.title('Spread between ' + tickers[2*i] + ' and ' + tickers[2*i+1])

png

Check For Cointegration

In this section, we test if two securities form a stationary or cointegrating pair. We use the augmented Dickey-Fuller (ADF) test where we have the following:

  1. The null hypothesis is that a unit root is present in the price series, it is non-stationary.
  2. The alternative is that unit root is not present in the prices series, it is stationary.

Run cointegration check using augmented Dickey-Fuller test

# Initialize stats
stats_list = []

for i in range(len(spread_list)):

    # ADF test
    stats = adfuller(spread_list[i])

    # Update stats
    stats_list.append(stats)

Set the pairs

# Initialize pairs
pairs = []

for i in range(no_pairs):
    # Update pairs
    pairs.append(tickers[2*i] + '/' + tickers[2*i+1])

Create stats DataFrame

# Initialize dict
stats_dict = {}

for i in range(no_pairs):

    # Update dict
    stats_dict[pairs[i]] = [stats_list[i][0],
                            stats_list[i][1],
                            stats_list[i][4]['1%'], stats_list[i][4]['5%'], stats_list[i][4]['10%']]

# Create DataFrame
stats_df = pd.DataFrame(stats_dict,
                          index=['ADF Statistic', 'P-value', '1%', '5%', '10%'])

Inspect

stats_df
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GDX/GLD MCD/YUM BTC-USD/ETH-USD
ADF Statistic -3.386075 -3.072452 -2.161601
P-value 0.011443 0.028660 0.220476
1% -3.448344 -3.447815 -3.448344
5% -2.869469 -2.869237 -2.869469
10% -2.570994 -2.570870 -2.570994

Remarks:

  1. For the spread between GDX and GLD, the ADF statistic is -3.39 which is lower than the 1% critical value -3.45, which means that there is a better than 99% probability that the spread between GDX and GLD is stationary.
  2. For the spread between MCD and YUM, the ADF statistic is -3.07 is between the 1% critical value -3.45 and 5% critical value of -2.87, which means that there is a better than 95% probability that the spread between MCD and YUM is stationary.
  3. For the spread between BTC-USD and ETH-USD, the ADF statistic is -2.16 which is higher than the critical values, which means that the spread between BTC-USD and ETH-USD is not stationary.

Conclusion

In this article, we demonstrated how to form a a good cointegrating pair of securities. We used the OLS method to determine the hedge ratio between securities; and the ADF test to check for stationarity. The results suggest the following: cointegraing pairs could be formed within gold (GDX and GLD) and fast food securities (MCD and YUM); and cointegrating pairs could not be formed within cryptocurrencies (BTC-USD and ETH-USD).