Stationarity

Time series stability testing

Stationarity is a fundamental assumption in time series analysis—understanding when it holds and how to achieve it.

1 Abstract

A stationary time series has statistical properties (mean, variance, autocorrelation) that don’t change over time. Most financial prices are non-stationary, but returns are typically stationary. Testing for stationarity using the Augmented Dickey-Fuller (ADF) test is essential before applying many time series models.

2 Definition

A time series \(\{X_t\}\) is strictly stationary if the joint distribution of \((X_{t_1}, ..., X_{t_k})\) is identical to \((X_{t_1+h}, ..., X_{t_k+h})\) for all \(h\).

Weak stationarity (more practical) requires:

1. Constant mean: \(E[X_t] = \mu\) for all \(t\)
2. Constant variance: \(Var(X_t) = \sigma^2\) for all \(t\)
3. Autocovariance depends only on lag: \(Cov(X_t, X_{t+h}) = \gamma(h)\)

3 Why Stationarity Matters

• Predictability: Non-stationary series have unpredictable statistical properties
• Model validity: ARMA, GARCH models assume stationarity
• Spurious regression: Regressing non-stationary series produces misleading results
• Mean reversion: Stationary series revert to their mean; non-stationary series don’t

4 Unit Root and Random Walk

A random walk is a classic non-stationary process:

\[ X_t = X_{t-1} + \varepsilon_t \]

This has a unit root (coefficient = 1). The variance grows without bound over time:

\[ Var(X_t) = t \cdot \sigma_\varepsilon^2 \]

5 Augmented Dickey-Fuller Test

The ADF test checks for a unit root:

\[ \Delta X_t = \alpha + \beta t + \gamma X_{t-1} + \sum_{i=1}^{p} \delta_i \Delta X_{t-i} + \varepsilon_t \]

• Null hypothesis: \(\gamma = 0\) (unit root exists, non-stationary)
• Alternative: \(\gamma < 0\) (stationary)
• Decision: Reject null if test statistic < critical value

6 Compute (Python)

	Series	ADF Statistic	p-value	Critical 1%	Critical 5%	Stationary
0	Prices	0.4934	0.9847	-3.4356	-2.8639	No
1	Log Returns	-22.0448	0.0000	-3.4356	-2.8639	Yes

7 Visual Comparison

8 Rolling Statistics

Non-stationary series have time-varying statistics.

9 Making Series Stationary

Common transformations:

1. Differencing: \(\Delta X_t = X_t - X_{t-1}\) (removes trend)
2. Log transform: Stabilizes variance
3. Seasonal differencing: For seasonal patterns
4. Detrending: Subtract fitted trend

	Series	ADF Statistic	p-value	Critical 1%	Critical 5%	Stationary
0	Prices (Level)	0.4934	0.9847	-3.4356	-2.8639	No
1	Prices (1st Diff)	-19.6391	0.0000	-3.4356	-2.8639	Yes
2	Prices (2nd Diff)	-13.6061	0.0000	-3.4357	-2.8639	Yes

10 Autocorrelation Analysis

Stationary series have autocorrelation that decays to zero; non-stationary series have persistent autocorrelation.

11 Multiple Assets

	Series	ADF Statistic	p-value	Critical 1%	Critical 5%	Stationary
0	SPY Price	0.4943	0.9847	-3.4356	-2.8639	No
1	SPY Returns	-22.0447	0.0000	-3.4356	-2.8639	Yes
2	TLT Price	-1.9229	0.3213	-3.4356	-2.8638	No
3	TLT Returns	-27.5126	0.0000	-3.4356	-2.8638	Yes
4	GLD Price	4.0465	1.0000	-3.4356	-2.8639	No
5	GLD Returns	-35.8817	0.0000	-3.4356	-2.8638	Yes
6	BTC-USD Price	-0.8090	0.8165	-3.4339	-2.8631	No
7	BTC-USD Returns	-8.0555	0.0000	-3.4340	-2.8631	Yes

12 Conclusion

Stationarity testing is a critical first step in time series analysis. Financial prices are typically non-stationary (trending, with growing variance), while returns are usually stationary. The ADF test provides a formal statistical test, and transformations like differencing or log returns can convert non-stationary series to stationary ones suitable for modeling.