7.3 Non-Stationary Time Series

Non-Stationary Time Series

Many risk-relevant series—price levels, cumulative P&L, some macro factors—are non-stationary. Applying stationary ARMA formulas without transformation produces spurious regressions, misleading R², and invalid t-stats. This section separates deterministic trends and seasonal means from stochastic unit-root trends and shows how forecasts change.

Trends: Deterministic Versus Stochastic

A deterministic trend example:

yₜ = α + β t + uₜ,

with uₜ stationary. The mean E[yₜ] = α + β t moves with calendar time, so {yₜ} is not covariance stationary, but demeaning by subtracting the fitted trend (or including t in a regression) can restore a stationary residual.

A stochastic trend (unit root) example is the random walk below: there is no fixed attractor; the “level” wanders. Differencing once, Δyₜ = yₜ − yₜ₋₁, often yields a stationary series (integrated of order 1, or I(1)).

Mis-detrending a unit root (treating it as deterministic) or over-differencing a trend-stationary series both distort inference—exam questions often probe which transformation matches which DGP.

Seasonal Dummies

Deterministic seasonality shifts the mean by season:

yₜ = α + Σ_{s=1}^{S−1} δₛ Dₛₜ + uₜ,

where Dₛₜ are seasonal dummy variables (omit one season if an intercept is present to avoid perfect collinearity). Quarterly S = 4; monthly S = 12.

Use seasonal dummies when the seasonal pattern is a stable calendar mean shift and residuals look stationary. If seasonality itself has a unit root (seasonal integration), seasonal differencing (1 − Lˢ) may be required instead of—or in addition to—dummies.

Worked dummy collinearity reminder

Monthly model with intercept plus 12 month dummies → perfect collinearity. Use intercept + 11 dummies, or 12 dummies and no intercept. Same logic as regression diagnostics in QA–9.

Random Walks and Unit Roots

Random walk (with no drift):

yₜ = yₜ₋₁ + εₜ, εₜ white noise.

Then yₜ = y₀ + Σ_{j=1}^{t} εⱼ, so Var(yₜ) = t σ² if Var(ε) = σ² and y₀ fixed. There is no mean reversion: E[yₜ₊ₕ | yₜ] = yₜ. ACF of levels stays near 1 for many lags—classic nonstationary symptom.

Random walk with drift: yₜ = μ + yₜ₋₁ + εₜ combines a deterministic drift μ t with a stochastic trend.

Unit root means the AR polynomial has a root equal to 1: (1 − L) factor. AR(1) with φ = 1 is the leading case. Near-unit roots (φ = 0.99) look similar in short samples but are still mean-reverting in population—hence testing pain.

Worked variance growth

σ = 1, y₀ = 0. After t = 100 steps, Var(y₁₀₀) = 100 under a pure random walk. A stationary AR(1) with φ = 0.9 has Var(y) = 1/(1 − 0.81) ≈ 5.26 regardless of t (in the limit). Levels of I(1) series wander much more than stationary alternatives.

Unit-Root Testing Challenges

The Dickey–Fuller (DF) / Augmented DF (ADF) approach tests H₀: φ = 1 in Δyₜ = c + δ t + π yₜ₋₁ + lags of Δy + eₜ, with π = φ − 1. Under the null, the t-statistic on π does not follow a Student-t; use Dickey–Fuller critical values (more negative than usual t crits).

Challenges FRM candidates must respect:

1. Low power against stationary alternatives with φ near 1—failure to reject does not prove a unit root.
2. Specification of deterministic terms (none / constant / constant+trend) changes null and critical values; wrong choice misleads.
3. Structural breaks can mimic unit roots if ignored.
4. Sample size: financial daily samples are long in n but may still have slow mean reversion economically.

h-Step Forecasts with Seasonality

For a stationary seasonal ARMA, forecasts converge to the unconditional mean, but at seasonal horizons the path still reflects seasonal AR/MA weights before dying out.

For deterministic seasonal means, the h-step forecast should use the dummy (or seasonal factor) appropriate for date t+h:

ŷₜ₊ₕ = α̂ + δ̂_{season(t+h)} (+ any stationary dynamics forecast).

Example: if Q4 always runs +2 above the average of other quarters in a pure dummy model, a forecast landing in Q4 should include that +2 even at long horizons—deterministic seasonality does not fade, unlike stationary stochastic seasonality.

For a seasonal random walk (yₜ = yₜ₋ₛ + εₜ), the forecast is ŷₜ₊ₕ = yₜ₊ₕ₋ₛₖ copying the last same-season observation for horizons that are multiples of s—shocks persist in that seasonal slot.

Worked seasonal forecast

Quarterly sales index with only seasonal dummies, estimated means: Q1 = 100, Q2 = 110, Q3 = 105, Q4 = 120. From the end of Q2, the 1-step forecast (Q3) is 105; the 2-step (Q4) is 120; the 3-step (next Q1) is 100. Horizons do not shrink these deterministic seasonal gaps.

Trend Forecasts and Prediction Intervals

Deterministic trend forecast: ŷₜ₊ₕ = α̂ + β̂ (t+h). Interval width depends mainly on residual variance and parameter uncertainty; if uₜ is stationary, interval width approaches a finite bound (plus parameter uncertainty) rather than growing like √h without limit.

Random-walk forecast: ŷₜ₊ₕ = yₜ (no drift) or yₜ + μ̂ h (with drift). Forecast-error variance ≈ h σ², so interval width grows like √h. Long-horizon fan charts for I(1) prices are inherently wide—critical for long-run risk communications.

Worked interval growth

Random walk, σ = 2, 95% approx. interval for h-step error using ±1.96 σ √h:

Uncertainty compounds with the square root of horizon under independent increments.

Practical FRM Checklist

1. Plot the series and ACF of levels vs differences.
2. Decide deterministic trend / seasonal dummies vs unit root / seasonal difference using economics plus careful tests.
3. Avoid regressing independent I(1) levels on each other without cointegration theory.
4. Match forecast formulas to the chosen DGP: fading (stationary), copying seasons (seasonal RW), or extrapolating t (deterministic trend).
5. Always widen intervals with horizon for stochastic trends.

Non-stationarity is not a nuisance to ignore: it changes what “mean,” “shock,” and “forecast risk” mean in market and credit time series.