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Stationarity (ADF & KPSS)

Stationarity (ADF & KPSS Tests)

Definition

Core Statement

A time series is Stationary if its statistical properties (mean, variance, autocorrelation) do not change over time. Stationarity is a prerequisite for many time series models like ARIMA Models.

Purpose

  1. Determine if a time series can be modeled with standard techniques.
  2. Decide if differencing or other transformations are needed.

When to Use

Test for Stationarity When...

  • Preparing data for ARIMA, VAR, or Granger Causality.
  • Visual inspection of the series shows a trend or changing variance.

Theoretical Background

Types of Non-Stationarity

Type Description Solution
Trend Mean changes over time. Differencing (ΔYt=Yt−Yt−1).
Changing Variance Volatility changes over time. Log transform; GARCH models.
Seasonality Periodic patterns. Seasonal differencing.

The Tests

Test Null Hypothesis (H0) Interpretation
ADF (Augmented Dickey-Fuller) Series has a unit root (Non-Stationary). Reject if p < 0.05: Series is stationary.
KPSS Series is stationary. Reject if p < 0.05: Series is non-stationary.
Use Both Tests Together

ADF Result KPSS Result Conclusion
Reject (Stationary) Fail to Reject (Stationary) Stationary.
Fail to Reject Reject (Non-Stationary) Non-Stationary. Difference needed.
Conflict Conflict Trend-stationary or need more investigation.

Worked Example: Is the Trend Real?

Problem

You have a stock price series Pt that looks like it's going up.

  • ADF p-value: 0.65
  • KPSS p-value: 0.01

Question: Is the series stationary? Should you difference it?

Analysis:

  1. ADF Test (H0: Non-Stationary):

    • p=0.65>0.05. Fail to reject H0.
    • Evidence suggests Non-Stationary.

  2. KPSS Test (H0: Stationary):

    • p=0.01<0.05. Reject H0.
    • Evidence suggests Non-Stationary.

  3. Conclusion:

    • Both tests agree. The series has a Unit Root.
    • Action: Apply First Difference (ΔPt=Pt−Pt−1).

  4. Re-Test (After Differencing):

    • New ADF p<0.01. New KPSS p>0.10.
    • Now it is stationary.

Assumptions

Limitations

Pitfalls

  1. Trend Stationarity vs Difference Stationarity: ADF assumes a stochastic trend suitable for differencing. If the trend is deterministic (e.g., a straight line y=mx+b), differencing introduces overdifferencing artifacts.
  2. Structural Breaks: A sudden crash (like 2008 financial crisis) is often interpreted by ADF as non-stationarity. Using standard ADF here is wrong; use Zivot-Andrews Test instead.
  3. Seasonality: A strong seasonal cycle can look like non-stationarity. Remove seasonality before testing.

Python Implementation

from statsmodels.tsa.stattools import adfuller, kpss

# ADF Test
adf_result = adfuller(series, autolag='AIC')
print(f"ADF Statistic: {adf_result[0]:.4f}")
print(f"ADF p-value: {adf_result[1]:.4f}")

# KPSS Test
kpss_result = kpss(series, regression='c')
print(f"KPSS Statistic: {kpss_result[0]:.4f}")
print(f"KPSS p-value: {kpss_result[1]:.4f}")

# Interpretation
if adf_result[1] < 0.05 and kpss_result[1] > 0.05:
    print("Series is STATIONARY.")
elif adf_result[1] > 0.05 and kpss_result[1] < 0.05:
    print("Series is NON-STATIONARY. Apply differencing.")

R Implementation

library(tseries)

# ADF Test
adf.test(series)

# KPSS Test
kpss.test(series, null = "Level")

# Interpretation:
# ADF p < 0.05 AND KPSS p > 0.05 -> Stationary.

Interpretation Guide

Test p-value Meaning
Test p-value Meaning
------ --------- ---------
ADF 0.02 Reject H0: Stationary.
ADF 0.35 Fail to reject: Unit Root (Non-Stationary).
KPSS 0.10 Fail to reject: Stationary.
KPSS 0.01 Reject: Non-Stationary.
Conflicting ADF > 0.05 & KPSS < 0.05 Non-Stationary. Difference it.
Conflicting ADF < 0.05 & KPSS > 0.05 Stationary. Use as is.