Stationarity Testing for Crypto Returns

1. What is Stationarity?

At its core, a time series is stationary if its statistical properties do not change over time. This is a crucial assumption for many time-series forecasting models, as they assume that the statistical patterns observed in the past will hold in the future.

Let's start with a formal definition and some visual intuition.

To be more precise, a time series $X_t$ is weakly stationary (or wide-sense stationary) if it satisfies three conditions:

1. Constant Mean: The expected value is constant for all $t$.
   $\mathbb{E}[X_t]=\mu$
2. Constant Variance: The variance is finite and constant for all $t$.
   $\text{Var}(X_t)={\sigma }^2<\infty$
3. Constant Autocovariance: The covariance between any two points in the series depends only on the time difference (the lag $k$) between them, not on their absolute position in time.
   $\text{Cov}(X_t,X_{t-k})={\gamma }_k$

As the video highlights, descriptive statistics like the overall mean are meaningless for a non-stationary series because they don't represent a stable underlying property. This is why we can't apply simple statistical models directly to asset prices, which typically exhibit strong trends.

2. From Non-Stationary Prices to Stationary Returns

Asset prices are classic examples of non-stationary time series. They tend to drift over time, a behavior often modeled as a random walk. A simple random walk can be expressed as:
$P_t=P_{t-1}+{ϵ}_t$
where ${ϵ}_t$ is a white noise term. This process has a "unit root" and is non-stationary; its variance grows with time.

To build valid statistical models, we need to transform this non-stationary price series into a stationary one. The most common method is differencing.

As we discussed in a previous lesson, calculating logarithmic returns is a standard practice:
$r_t=\log (P_t)-\log (P_{t-1})$
This is the first difference of the log-price series. If log-prices follow a random walk (are I(1)), their first difference—the log returns—should be stationary (I(0)). This is a foundational assumption in financial econometrics.

Our goal now is to move beyond visual inspection and formally test this hypothesis on our crypto data.

3. Statistical Tests for Stationarity

Visual inspection can be misleading. We need objective statistical tests to determine if a series is stationary. We will focus on two of the most widely used tests: the Augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. They are powerful when used in combination because they have opposing null hypotheses.

Here's a summary of the tests:

Notice the reversal. A low p-value in an ADF test suggests stationarity, while a low p-value in a KPSS test suggests non-stationarity. This is why using them together is so effective.

4. Interpreting Results for Strategy Selection

By running both tests, we can diagnose the nature of a time series with more confidence. This directly impacts our strategy choice. For instance, a momentum strategy is predicated on the existence of a trend, which is a feature of a non-stationary series.

Here is the interpretation matrix based on the article:

1. ADF: Stationary & KPSS: Stationary

   • Conclusion: The series is stationary.
   • Strategy Implication: The series is mean-reverting. This is a good candidate for strategies that trade reversals (e.g., buy dips, sell rips). We don't expect to find this in raw price data but might find it in the returns of a stable asset or the spread between two cointegrated assets.

2. ADF: Non-stationary & KPSS: Non-stationary

   • Conclusion: The series is non-stationary.
   • Strategy Implication: The series is not mean-reverting and likely contains a stochastic trend (like a random walk). This is the behavior we expect from asset prices and is the domain of momentum and trend-following strategies.

3. ADF: Non-stationary & KPSS: Stationary

   • Conclusion: The series is trend-stationary. It has a deterministic trend. If you were to subtract this trend, the remaining series would be stationary.
   • Strategy Implication: This is also a candidate for trend-following, but the nature of the trend is more predictable than a purely stochastic one.

4. ADF: Stationary & KPSS: Non-stationary

   • Conclusion: The series is difference-stationary. It is I(1).
   • Strategy Implication: This confirms the random walk-like behavior. Differencing the series once will make it stationary. This is the classic signature of asset prices, where the price itself is unpredictable, but the returns are stationary and more amenable to modeling.

5. Practical Implementation with Python

Let's apply these tests to real crypto data. We will test both the closing prices and the log returns of a crypto asset. We expect to see prices being non-stationary (Case 2 or 4) and returns being stationary (Case 1).

The statsmodels library in Python provides implementations for both tests.

import pandas as pd
import numpy as np
from statsmodels.tsa.stattools import adfuller, kpss

def analyze_stationarity(timeseries, series_name=""):
    """
    Performs and interprets ADF and KPSS tests for stationarity.

    Args:
        timeseries (pd.Series): The time series data to analyze.
        series_name (str): A name for the series for printing.
    """
    print(f"--- Stationarity Analysis for {series_name} ---")
    
    # --- ADF Test ---
    # H0: Series is non-stationary
    adf_result = adfuller(timeseries.dropna())
    adf_p_value = adf_result[1]
    
    print(f'ADF Statistic: {adf_result[0]:.4f}')
    print(f'ADF p-value: {adf_p_value:.4f}')
    
    if adf_p_value <= 0.05:
        print("ADF Conclusion: Reject H0. The series is likely stationary.")
        adf_stationary = True
    else:
        print("ADF Conclusion: Fail to reject H0. The series is likely non-stationary.")
        adf_stationary = False

    print("-" * 20)

    # --- KPSS Test ---
    # H0: Series is stationary
    # Using 'c' for constant-only trend. Use 'ct' for constant and trend.
    kpss_result = kpss(timeseries.dropna(), regression='c', nlags='auto')
    kpss_p_value = kpss_result[1]

    print(f'KPSS Statistic: {kpss_result[0]:.4f}')
    print(f'KPSS p-value: {kpss_p_value:.4f}')
    # Note: The p-value can be 0.10 (upper bound) or 0.01 (lower bound).
    # We check against our significance level.
    
    if kpss_p_value <= 0.05:
        print("KPSS Conclusion: Reject H0. The series is likely non-stationary.")
        kpss_stationary = False
    else:
        print("KPSS Conclusion: Fail to reject H0. The series is likely stationary.")
        kpss_stationary = True
        
    print("-" * 20)

    # --- Combined Conclusion ---
    if adf_stationary and kpss_stationary:
        print("Combined Conclusion: The series is stationary (Case 1).")
        print("Implication: Suitable for mean-reversion strategies.")
    elif not adf_stationary and not kpss_stationary:
        print("Combined Conclusion: The series is non-stationary (Case 2).")
        print("Implication: Suitable for trend-following/momentum strategies.")
    elif not adf_stationary and kpss_stationary:
        print("Combined Conclusion: The series is trend-stationary (Case 3).")
        print("Implication: A deterministic trend exists. Suitable for trend-following.")
    elif adf_stationary and not kpss_stationary:
        print("Combined Conclusion: The series is difference-stationary (Case 4).")
        print("Implication: Requires differencing to become stationary. Classic price behavior.")
    
    print("-" * 40 + "\n")


# --- Example Usage ---
# Assume you have a DataFrame 'df' with OHLCV data and a datetime index
# For demonstration, let's create some sample data
np.random.seed(42)
price_drift = 0.0005
price_vol = 0.02
prices = 100 * np.exp(np.cumsum(np.random.normal(price_drift, price_vol, 1000)))
dates = pd.to_datetime(pd.date_range(start='2021-01-01', periods=1000))
df = pd.DataFrame({'close': prices}, index=dates)

# Calculate log returns
df['log_return'] = np.log(df['close']).diff()

# Analyze the closing price series
analyze_stationarity(df['close'], "BTC-USD Closing Price (Simulated)")

# Analyze the log return series
analyze_stationarity(df['log_return'], "BTC-USD Log Returns (Simulated)")

When you run this code, you should see that the price series is identified as non-stationary, justifying the use of momentum strategies. The log return series, in contrast, will be identified as stationary, confirming it's a valid input for statistical models that require this property.

Conclusion

Key Takeaways:

• Stationarity implies that a time series has a constant mean, variance, and autocovariance structure.
• Asset prices are typically non-stationary (often I(1) or "difference-stationary"), exhibiting trends that make them unsuitable for many standard statistical models.
• Asset returns (especially log returns) are often stationary (I(0)), making them the primary object of study for quantitative modeling.
• The Augmented Dickey-Fuller (ADF) test has a null hypothesis of non-stationarity.
• The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test has a null hypothesis of stationarity.
• By combining the results of both tests, we can confidently diagnose a series and select an appropriate strategy class: momentum for non-stationary series and mean-reversion for stationary series.

You now have the theoretical understanding and practical tools to assess the most fundamental property of your trading data. This is a critical step before we move on to feature engineering and signal generation.

Preview of the Next Lesson:

Our analysis so far has assumed clean, continuous data. However, real-world crypto data is messy. It contains gaps from exchange downtime, sudden anomalous spikes from flash crashes, and periods of zero volume. Before we can reliably calculate signals, we must address these issues. In the next lesson, we will focus on the learning outcome: "Implement robust methods to detect and handle crypto-specific data anomalies, such as exchange downtime, flash crashes, and missing values."