Stationarity in Time Series — Tests and Transformations

Statistics

Making Time Series Ready for Modeling

Stationarity — constant mean, variance, and autocorrelation over time — is a prerequisite for most forecasting models. ADF and KPSS tests detect non-stationarity, while differencing and transformations restore it.

Economic Forecasting — Transform non-stationary GDP data for reliable ARIMA modeling
Financial Analysis — Convert price series to stationary returns for volatility modeling
Climate Science — Remove trends from temperature data to analyze cyclical patterns

Stationary series are predictable because their statistical properties don't wander.

A time series is stationary if its statistical properties (mean, variance, autocorrelation) do not change over time. Stationarity is a key assumption for many time series models.

DfStationary Time Series

A time series $\{Y_t\}$ is strictly stationary if the joint distribution of $(Y_{t_1}, Y_{t_2}, \ldots, Y_{t_k})$ is the same as $(Y_{t_1+\tau}, Y_{t_2+\tau}, \ldots, Y_{t_k+\tau})$ for all $k$ , all time indices, and all lags $\tau$ .

DfWeakly Stationary

A time series is weakly (covariance) stationary if:

$E[Y_t] = \mu$ (constant mean)
$\text{Var}(Y_t) = \sigma^2$ (constant variance)
$\text{Cov}(Y_t, Y_{t+h}) = \gamma(h)$ (autocovariance depends only on lag h)

Why Stationarity Matters

Key Insight

Most time series models (AR, MA, ARIMA) are built on the assumption of stationarity. If the series is non-stationary, parameter estimates may be unreliable and forecasts may be misleading.

Non-stationary series often exhibit:

Trend: systematic increase or decrease in mean
Seasonality: periodic fluctuations
Heteroscedasticity: changing variance over time

Augmented Dickey-Fuller (ADF) Test

The ADF test checks for a unit root, which indicates non-stationarity.

ADF Test Regression

\Delta Y_t = \alpha + \beta t + \gamma Y_{t-1} + \sum_{i=1}^{p} \delta_i \Delta Y_{t-i} + \varepsilon_t

Here,

$\Delta Y_t$ =First difference of Y at time t
$\gamma$ =Coefficient on lagged level (key parameter)
$\alpha$ =Constant (intercept)
$\beta t$ =Time trend
$p$ =Number of lagged difference terms

| Hypothesis | Meaning |

|-----------|---------|

| $H_0$ : $\gamma = 0$ | Unit root present -> non-stationary |

| $H_1$ : $\gamma < 0$ | No unit root -> stationary |

Interpreting ADF

If p-value < 0.05, reject $H_0$ and conclude the series is stationary. If p-value > 0.05, fail to reject — the series may be non-stationary.

KPSS Test

The KPSS test has opposite hypotheses from the ADF test.

KPSS Test

Y_t = \xi_t + u_t, \quad \xi_t = \xi_{t-1} + \varepsilon_t

Here,

$\xi_t$ =Random walk component
$u_t$ =Stationary error
$\varepsilon_t$ =White noise innovation

| Hypothesis | Meaning |

|-----------|---------|

| $H_0$ : Series is stationary | Trend-stationary |

| $H_1$ : Series is non-stationary | Unit root present |

Using Both Tests

Use ADF and KPSS together. If ADF rejects $H_0$ (stationary) and KPSS fails to reject $H_0$ (stationary), the series is likely stationary. If both reject, the series is borderline.

Transformations for Stationarity

Differencing

First Differencing

\Delta Y_t = Y_t - Y_{t-1}

Here,

$\Delta Y_t$ =First difference — removes linear trend
$Y_t$ =Value at time t

For seasonal data, use seasonal differencing:

Seasonal Differencing

\Delta_s Y_t = Y_t - Y_{t-s}

Here,

$s$ =Seasonal period (e.g., 12 for monthly data)

Log Transformation

Stabilizes variance when it increases with the level of the series.

Log Transformation

Z_t = \ln(Y_t)

Here,

$Y_t$ =Original positive time series
$Z_t$ =Transformed series with stabilized variance

Box-Cox Transformation

A family of power transformations that includes log as a special case.

Box-Cox Transformation

Y_t^{(\lambda)} = \begin{cases} \frac{Y_t^\lambda - 1}{\lambda} & \lambda \neq 0 \ \ln(Y_t) & \lambda = 0 \end{cases}

Here,

$\lambda$ =Transformation parameter (estimated from data)

Python Implementation


import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

from statsmodels.tsa.stattools import adfuller, kpss



np.random.seed(42)



# Non-stationary series (random walk + trend)

t = np.arange(200)

Y = np.cumsum(np.random.randn(200)) + 0.02 * t



# ADF test

adf_result = adfuller(Y, autolag='AIC')

print(f"ADF Statistic: {adf_result[0]:.3f}")

print(f"p-value: {adf_result[1]:.4f}")



# KPSS test

kpss_result = kpss(Y, regression='ct', nlags='auto')

print(f"KPSS Statistic: {kpss_result[0]:.3f}")

print(f"p-value: {kpss_result[1]:.4f}")



# Apply differencing

Y_diff = np.diff(Y)



adf_diff = adfuller(Y_diff, autolag='AIC')

print(f"\nAfter differencing:")

print(f"ADF p-value: {adf_diff[1]:.4f}")

Worked Example

Example: Testing Stationarity

A monthly airline passenger series shows increasing variance and trend.

Log transform to stabilize variance: $Z_t = \ln(Y_t)$
ADF test on $Z_t$ : p-value = 0.85 -> non-stationary
First difference: $\Delta Z_t = Z_t - Z_{t-1}$
ADF test on $\Delta Z_t$ : p-value = 0.001 -> stationary
Seasonal difference (s=12): $\Delta_{12} Z_t$
ADF test on $\Delta_{12} Z_t$ : p-value = 0.000 -> stationary

The series requires both seasonal and first differencing to achieve stationarity.

Key Takeaways

Summary: Stationarity

A stationary time series has constant mean, variance, and autocovariance over time
The ADF test checks for unit roots ( $H_0$ : non-stationary)
The KPSS test has opposite hypotheses ( $H_0$ : stationary)
Use both tests together for more reliable conclusions
Differencing removes trends; log transformation stabilizes variance
Most ARIMA models require a stationary input series

Stationarity in Time Series — Tests and Transformations

Stationarity in Time Series — Tests and Transformations

Making Time Series Ready for Modeling

DfStationary Time Series

DfWeakly Stationary

Why Stationarity Matters

Augmented Dickey-Fuller (ADF) Test

ADF Test Regression

KPSS Test

KPSS Test

Transformations for Stationarity

Differencing

First Differencing

Seasonal Differencing

Log Transformation

Log Transformation

Box-Cox Transformation

Box-Cox Transformation

Python Implementation

Worked Example

Example: Testing Stationarity

Key Takeaways

Summary: Stationarity

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