✅ Augmented Dickey-Fuller (ADF) Test

The ADF Test is a statistical test used to check whether a time series is stationary. It is one of the most commonly used methods in time series analysis.

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📌 Why is Stationarity Important?

Most statistical models like ARIMA, SARIMA, and even seasonal decomposition assume that the mean, variance, and autocorrelation structure of the time series is constant over time (i.e., the series is stationary).

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📐 What is the ADF Test?

The Augmented Dickey-Fuller Test tests the null hypothesis:

H₀: The time series has a unit root (non-stationary)

H₁: The time series is stationary

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📊 ADF Regression Model:

ADF runs a regression of the following form:

\[ \Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_p \Delta y_{t-p} + \varepsilon_t \]

Where:

• \(\Delta y_t\) is the first difference of the series
• \(\gamma\) is the coefficient tested against zero

If \(\gamma < 0\) and significantly different from zero, the series is stationary.

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✅ Interpreting ADF Test Results

• ADF Statistic: More negative → more likely to reject H₀

• p-value:

• If p < 0.05: Reject H₀ → the series is stationary

• If p > 0.05: Fail to reject H₀ → the series is non-stationary

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🧪 Code Example: Run ADF Test

from statsmodels.tsa.stattools import adfuller
import pandas as pd

# Load data
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
df = pd.read_csv(url, parse_dates=['Month'], index_col='Month')

# ADF Test
adf_result = adfuller(df['Passengers'])

# Output results
print("ADF Statistic: {:.4f}".format(adf_result[0]))
print("p-value: {:.4f}".format(adf_result[1]))
print("Critical Values:")
for key, value in adf_result[4].items():
    print(f"   {key}: {value:.4f}")

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⚠️ If the Series is Not Stationary

Use differencing or log transformation:

# First-order differencing
df['diff'] = df['Passengers'].diff()

# Drop NA and re-test
adf_result_diff = adfuller(df['diff'].dropna())

print("\nAfter Differencing:")
print("ADF Statistic:", adf_result_diff[0])
print("p-value:", adf_result_diff[1])

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📌 Summary Table

Metric	Description
ADF Statistic	Test statistic for checking stationarity
p-value	If < 0.05 → data is stationary
Critical Values	Thresholds at 1%, 5%, 10% levels
Differencing	Used to make non-stationary series stationary

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🧠 Bonus Tip: Visual Check of Stationarity

import matplotlib.pyplot as plt

# Visual check
df['diff'].dropna().plot(title='First Differenced Series')
plt.grid(True)
plt.show()