What is stationarity, why does it matter for risk models, and how do you test for it?

Stationarity is the foundation requirement for most time series models. A stationary process has statistical properties (mean, variance, autocorrelation) that don't change over time.

Types of Stationarity:

Strict stationarity: The entire joint distribution is invariant to time shifts. Too strong for practical use.

Weak (covariance) stationarity: Requires three conditions:

1. Constant mean: E(Y_t) = mu for all t
2. Constant variance: Var(Y_t) = sigma^2 for all t
3. Autocovariance depends only on lag: Cov(Y_t, Y_{t-k}) = gamma(k) for all t

What Breaks Without Stationarity:

• Regression is spurious: Two non-stationary series can show R^2 > 0.9 with zero causal relationship (e.g., GDP and stock prices both trend upward)
• Forecasts diverge: Forecasts from non-stationary models can drift without bound
• Test statistics are invalid: t-tests and F-tests have wrong distributions, leading to false significance
• VaR is meaningless: If the return distribution changes over time, yesterday's VaR says nothing about tomorrow

Unit Root Tests — Augmented Dickey-Fuller (ADF):

The ADF test checks whether a series has a unit root (non-stationary).

Test regression: Delta(Y_t) = alpha + betat + gammaY_{t-1} + Sum of delta_i*Delta(Y_{t-i}) + epsilon_t

• H0: gamma = 0 (unit root exists, non-stationary)
• H1: gamma < 0 (no unit root, stationary)

The test statistic = gamma / SE(gamma), compared to special Dickey-Fuller critical values (NOT standard t-distribution!).

Example — Foxworth Research:

Testing daily EURUSD exchange rates (1,000 observations):

• ADF test statistic: -1.87
• 5% critical value: -2.86
• Since -1.87 > -2.86, we FAIL to reject H0 -> exchange rate is non-stationary

Now test daily EURUSD returns (first differences):

• ADF test statistic: -28.4
• 5% critical value: -2.86
• Since -28.4 < -2.86, we REJECT H0 -> returns ARE stationary

Common Fix: First Differencing

Most financial prices are non-stationary (they wander), but returns (price changes) are stationary. This is why risk models use returns, not prices.

FRM Key Points:

• Prices are typically I(1) — integrated of order 1 — requiring one difference to achieve stationarity
• Use ADF or Phillips-Perron test to verify stationarity before fitting ARMA or GARCH models
• If two non-stationary series are cointegrated, their linear combination IS stationary (useful for pairs trading)
• ADF critical values are non-standard — don't use regular t-tables

Practice stationarity testing in our FRM Part I question bank.