Stationarity in Time Series

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What stationarity means

A time series is stationary if its statistical properties — mean, variance, and autocorrelation structure — don't change over time. Formally, a series $y_t$ is weakly (covariance) stationary if:

$E[y_t] = \mu$ for all $t$ (constant mean)
$\text{Var}(y_t) = \sigma^2$ for all $t$ (constant variance)
$\text{Cov}(y_t, y_{t+k})$ depends only on the lag $k$, not on $t$ itself (autocovariance depends only on lag)

Stationarity matters because most classical forecasting models — ARIMA, exponential smoothing in its simplest forms, and many statistical tests — are built on the assumption that the process generating the data is stable through time. A non-stationary series (one with a trend, a changing variance, or seasonal drift) will make these models produce unreliable forecasts and spurious statistical relationships.

What non-stationarity looks like

Three common sources of non-stationarity:

Trend: the mean drifts up or down over time (e.g. GDP growing year over year).

Seasonality: the mean or variance cycles predictably (e.g. retail sales spiking every December).

Changing variance (heteroskedasticity): the spread of the series widens or narrows over time (e.g. stock returns becoming more volatile during a crisis).

A classic trap is spurious correlation: two unrelated non-stationary series (say, cumulative rainfall in one country and a stock index in another) can show a strong correlation purely because they both trend upward, with no real causal link. This is why checking stationarity — or at least understanding why it's violated — comes before fitting most time series models, not after.

Augmented Dickey-Fuller test for stationarity

python

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

# series: a pandas Series of observations ordered by time
result = adfuller(series.dropna())

adf_statistic, p_value, used_lag, n_obs, critical_values, icbest = result
print(f"ADF statistic: {adf_statistic:.4f}")
print(f"p-value: {p_value:.4f}")
print("Critical values:", critical_values)

if p_value < 0.05:
    print("Reject null hypothesis -> series is likely stationary")
else:
    print("Fail to reject null -> series likely has a unit root (non-stationary)")

The Augmented Dickey-Fuller (ADF) test

The ADF test checks for a unit root — a specific mathematical signature of non-stationarity where shocks to the series persist forever rather than decaying. The null hypothesis is that a unit root is present (non-stationary); a small p-value (typically < 0.05) lets you reject that null and conclude the series is stationary. It's the most widely used stationarity test, though the KPSS test is often run alongside it because it has the opposite null hypothesis (stationarity), and agreement between the two gives more confidence than either alone.

Making a series stationary: differencing and transforms

The most common fix for a trending series is differencing: instead of modeling $y_t$ directly, model the change $y_t - y_{t-1}$. This removes a linear trend because a constant upward drift becomes a roughly constant difference. If one differencing pass isn't enough, a second difference, $(y_t - y_{t-1}) - (y_{t-1} - y_{t-2})$, can remove quadratic trends — though differencing more than twice is rarely needed and starts to amplify noise.

For seasonality, seasonal differencing subtracts the value from one full seasonal period earlier: $y_t - y_{t-m}$, where $m$ is the season length (12 for monthly data with yearly seasonality). For changing variance, a log transform (or Box-Cox transform more generally) often stabilizes variance before differencing is applied to the mean.

Differencing and re-testing stationarity

python

import numpy as np
from statsmodels.tsa.stattools import adfuller

def adf_pvalue(s):
    return adfuller(s.dropna())[1]

print("Original series p-value:", adf_pvalue(series))

# First difference removes a linear trend
diff1 = series.diff()
print("First difference p-value:", adf_pvalue(diff1))

# Seasonal difference (e.g. monthly data, yearly seasonality -> lag 12)
seasonal_diff = series.diff(12)
print("Seasonal difference p-value:", adf_pvalue(seasonal_diff))

# Log transform to stabilize variance before differencing the mean
log_diff = np.log(series).diff()
print("Log-differenced p-value:", adf_pvalue(log_diff))

Over-differencing has a cost

Differencing is not free — each pass discards one observation and can introduce artificial negative autocorrelation if applied beyond what's needed. A series that's already stationary, when differenced again, often shows a strong negative correlation at lag 1 purely as an artifact of the operation. Always re-check stationarity after each differencing pass rather than differencing repeatedly 'just in case.'

What's Next?

Once a series has been made stationary — through differencing, transforms, or both — it becomes suitable input for the classical forecasting models built around the stationarity assumption. The most widely used of these combines autoregression, differencing, and a moving-average error structure into a single framework worth understanding in detail.

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