Testing stationary process and time-series in Python (using cryptos)

• ADF and KPSS tests for checking stationarity;
• Obtaining market data of cryptocurrencies;
• Discussion on statistical tests;

Hello, all!

It is important to check stationarity!

In the analysis of a time-series, such as asset prices, one important characteristic to evaluate is the presence of a stationary behavior. In a stationary process, the values of the time-series, even stochastic, follows a trend towards a determined mean. Notice that a test statistic to verify stationarity is crucial, once that the fitting of recursive models can be simply meaningless even if the $R^2$ is high.
In quantitative finance, for instance, mean-reverting models assume that determined parameter (e.g. prices, volatility) follows a stationary process. In the following, I present a code used to generate a typical stationary process by means of a Ornstein-Uhlenbach model:

$dx=  (\theta - \alpha x)dt + \sigma dW$

where $\theta$ is the long-term average and $W$ denotes a Brownian motion. Hence:

	from math import sqrt
	import numpy.random as nr
	import matplotlib.pyplot as plt
	nr.seed(1314)
	#parameters
	sigma = 60
	alpha = 50
	beta = 100
	S0 = 90
	days = [d for d in range(90)]
	# contructing the random path
	def mean_reverting(days, alpha, beta, sigma):
	dW = nr.standard_normal(len(days))
	time_series=[]
	for d in range(len(days)):
	if d!=0:
	time_series.append(vec_S[d-1]+alpha*(beta-vec_S[d-1])*1/365+sigma*sqrt(1/365)*dW[d-1])
	else:
	time_series.append(S0)
	return time_series
	time_series = mean_reverting(days, alpha, beta, sigma)
	#plot
	def plot_(x, y, beta, label='Asset', label_vert_line='long-term average'):
	plt.plot(x, y, label=label)
	plt.axhline(beta, color='red', linestyle='dashed', label=label_vert_line)
	plt.ylabel('Price')
	plt.xlabel('Days')
	plt.legend(loc='best')
	plot_(days, time_series, beta)

view raw mean_reverting.py hosted with ❤ by GitHub

Notice that the series tends to return to the mean value. Of course, this is a very interesting characteristic for trades, for example, since one has an indication of the current average adopted by the time series. However, one should always remember that the analysis presented hereafter (as well as any other statistical model) does not predict a value in the future. This is because the current regime can change at any point in time, modifying completely the parameters of the stochastic model.

Getting crypto market data

In order to show the application of the stationary tests, I will use market data of three important cryptocurrencies: Bitcoin, Ethereum, and TrueUSD. I will use the API cryptoCMD which connects to the site CoinMarketCap to obtain the data we will evaluate:

	import cryptocmd
	from cryptocmd import CmcScraper
	def getCoinData(list_coin,numeraire,date_min,date_max):
	coin_data={}
	coin_name=[]
	if numeraire!='':
	try:
	scrapper = CmcScraper(numeraire, date_min, date_max)
	numeraire_data = scrapper.get_dataframe()
	except Exception as e:
	print("[CryptoDataParse - Numeraire]: type error: " + str(e))
	else:
	numeraire_data=[]
	for coin in list_coin:
	if coin != numeraire:
	try:
	scrapper=CmcScraper(coin,date_min, date_max)
	coin_data[coin]=scrapper.get_dataframe()
	coin_name.append(coin)
	except Exception as e:
	print("[CryptoDataParse]: type error: " + str(e))
	return coin_data, numeraire_data, coin_name

view raw Get_crypto_data.py hosted with ❤ by GitHub

The variable 'numeraire' I have created in order to take into account the possibility of getting the coin values as function of a third one (like LTC, for instance) instead of dollar.

Now that we already have the data we need, let´s move forward to the stationary tests.

Test statistics

I will show two different statistical tests. The first is the Augmented Dick-Fuller test (ADF) whose null hypothesis is the presence of a unit root in the sample of the time-series against the alternative of stationarity or trend-stationarity. The second test is the Kwiatkowski-Phillips-Schmidt-Shin test (KPSS) whose null hypothesis is the stationarity (or trend-stationarity) against the alternative of the presence of unit root. In fact, as recommended in the must-read paper of Kwiatkowski et al. published in 1992, "Testing the null hypothesis of stationarity against alternative of unit root", one should perform both kinds of test in order to fully characterize a time-series. In the case of the methods chosen in this post, a high likelihood of stationarity in the sample of a time-series will outcome from the cases in which null hypothesis is rejected in the ADF test but accepted in the KPSS test.

Quote from Kwiatkowski´s paper.

ADF test

The ADF in Python is within the powerful package statsmodel and the implementation is quite easy:

	from statsmodels.tsa.stattools import adfuller
	def perform_ADF_test(X):
	ADF_test = adfuller(X)
	print('ADF Statistic: %f' % ADF_test[0])
	print('p-value: %f' % ADF_test[1])
	print('Critical Values:')
	for key, value in ADF_test[4].items():
	print('\t%s: %.3f' % (key, value))
	perform_ADF_test(vec_S)

view raw ADF_test.py hosted with ❤ by GitHub

Let´s us take a look to the results of each of the coins chosen before [BTC, ETH, TUSD]:

	Coin: BTC
	ADF Statistic: -0.822876
	p-value: 0.812310
	Critical Values:
	1%: -3.454
	5%: -2.872
	10%: -2.572
	Coin: ETH
	ADF Statistic: -1.447629
	p-value: 0.559242
	Critical Values:
	1%: -3.456
	5%: -2.873
	10%: -2.573
	Coin: TUSD
	ADF Statistic: -4.618161
	p-value: 0.000120
	Critical Values:
	1%: -3.462
	5%: -2.876
	10%: -2.574

view raw ADF.txt hosted with ❤ by GitHub

Note that for both BTC and ETH the value of the ADF statistic is considerably higher than the critical values, even considering the higher tolerance 10%. In addition, the p-value is high. These factors indicate that for both BTC and ETH the null hypothesis of the presence of unit root should not be reject. Conversely, the results for TUSD show lower ADF statistics compared to the critical values as well as a low p-value, both indicating that we should reject the hypothesis of a unit root. 

Intuitively, the result is coherent, considering that BTC and ETH behaves more similarly to random walks whereas TUSD oscillates around the value of one dollar as initially proposed in the corresponding white paper.

KPSS test

Using KPSS we directly check the null hypothesis of stationarity in the time-series, which allows to complement the indication of the ADF test. The implementation is simple:

	from statsmodels.tsa.stattools import kpss
	def perform_kpss_test(X):
	kpss_test = kpss(X)
	print('KPSS Statistic: %f' % kpss_test[0])
	print('p-value: %f' % kpss_test[1])
	print('lags: %f' % kpss_test[2])
	print('Critical Values:')
	for key, value in kpss_test[3].items():
	print('\t%s: %.3f' % (key, value))

view raw kpss_test.py hosted with ❤ by GitHub

providing us the following results:

	Coin: BTC
	KPSS Statistic: 1.219533
	p-value: 0.010000
	lags: 16.000000
	Critical Values:
	10%: 0.347
	5%: 0.463
	2.5%: 0.574
	1%: 0.739
	Coin: ETH
	KPSS Statistic: 1.349860
	p-value: 0.010000
	lags: 16.000000
	Critical Values:
	10%: 0.347
	5%: 0.463
	2.5%: 0.574
	1%: 0.739
	Coin: TUSD
	KPSS Statistic: 0.127700
	p-value: 0.100000
	lags: 15.000000
	Critical Values:
	10%: 0.347
	5%: 0.463
	2.5%: 0.574
	1%: 0.739

view raw kpss_example.txt hosted with ❤ by GitHub

Once again, for both BTC and ETH the statistics of the test is higher than the critical values and for TUSD the kpss statistics is lower than the critical values. These results lead us to accept the null hypothesis of stationarity for the TUSD and reject it for BTC and ETH. In order to be more certain about our analysis, I can also test BTC and ETH once again, this time with respect to the possibility of trend-stationarity, in which stationarity might be related to a mean value which decreases (or increases) linearly with time. This can be done by simply introducing the argument regression='ct' in the kpss test and the results are presented below:

	Coin: BTC
	KPSS Statistic: 0.186164
	p-value: 0.021188
	lags: 16.000000
	Critical Values:
	10%: 0.119
	5%: 0.146
	2.5%: 0.176
	1%: 0.216
	Coin: ETH
	KPSS Statistic: 0.136235
	p-value: 0.068084
	lags: 16.000000
	Critical Values:
	10%: 0.119
	5%: 0.146
	2.5%: 0.176
	1%: 0.216
	Coin: TUSD
	KPSS Statistic: 0.082340
	p-value: 0.100000
	lags: 15.000000
	Critical Values:
	10%: 0.119
	5%: 0.146
	2.5%: 0.176
	1%: 0.216

view raw kpss_ct_results.txt hosted with ❤ by GitHub

For TUSD the results remain consistent, confirming by far the stationarity. For BTC, under the a typical alpha value of 5%, the low p-value for this coin suggests the rejection of the null hypothesis for this case. For ETH, the p-value of 6.8% is higher than the typical alpha of 5%, then we compare the kpss statistic against the critical values. In fact, we can only reject the null hypothesis under the highest critical value (at 10%), while using more relaxing criteria we cannot reject the hypothesis of trend stationarity for the coin ETH.

In the example present above, the results of the ADF and KPSS complement themselves allowing to us to characterize the time-series with respect to unit root and stationarity. As mentioned before, stationarity is a worthful property for quantitative trading strategies. It is worth mentioning that even if a determined asset does not behaves suitably to the strategy, one can always try different portfolios that might behave as a stationary process, even for a short duration.

I hope you enjoyed the post. Leave your comments and share.

May the Force be with you.

#=================================
Diogo de Moura Pedroso
LinkedIn: www.linkedin.com/in/diogomourapedroso