Nonstationary Time Series: formula sheet

Key formulas

Constant mean

E[Y(t)] = μ for all t

First condition of covariance stationarity.

Constant, finite variance

Var[Y(t)] = σ² < ∞ for all t

Second condition.

Autocovariance depends only on lag

Cov[Y(t), Y(t-k)] = γ(k), independent of t

Third condition. γ(0) equals the variance.

Autocorrelation

ρ(k) = γ(k) ÷ γ(0)

For a stationary series, ρ(k) typically decays toward zero as k grows.

Random walk (nonstationary)

Y(t) = Y(t-1) + e(t), so Var[Y(t)] = tσ² (starting from a fixed Y(0))

Variance grows with t, so it fails the constant-variance condition.

Deterministic trend

Y(t) = a + bt + e(t), E[Y(t)] = a + bt

Mean depends on t when b ≠ 0. Detrending gives a stationary series.

Linear trend

y_t = β0 + β1·t + ε_t

β1 is the constant change in y per period. t = 1, 2, ..., T.

Quadratic trend

y_t = β0 + β1·t + β2·t² + ε_t

Slope at time t = β1 + 2β2·t. A turning point exists at t = −β1 ÷ (2β2) when β2 ≠ 0.

Log-linear (exponential) trend

ln y_t = β0 + β1·t + ε_t, equivalently y_t = exp(β0 + β1·t + ε_t)

Requires y_t > 0. β1 is the continuously compounded growth rate per period.

Exact growth rate from log-linear model

Growth per period = exp(β1) − 1

For small β1 this is close to β1. Over n periods, growth = exp(n·β1) − 1.

Trend forecast, h periods ahead

Linear: ŷ_{T+h} = β̂0 + β̂1·(T+h). Log-linear: ŷ_{T+h} = exp(β̂0 + β̂1·(T+h))

Plug the future time index into the fitted equation. Ignore the error term, whose expected value is zero.

Detrended series (trend stationary)

Residual_t = y_t − fitted trend_t

If the series is trend stationary, the residuals are covariance stationary and can be modelled with AR/MA terms.

AR(1) model

Y(t) = δ + φY(t-1) + ε(t)

Stationary if |φ| < 1. Unit root if φ = 1.

Random walk

Y(t) = Y(t-1) + ε(t)

ε is white noise with variance σ². Shocks have permanent effects.

Random walk with drift

Y(t) = δ + Y(t-1) + ε(t)

δ is the expected change per period.

Backward substitution (with drift)

Y(t) = Y(0) + δt + Σ ε(i), i = 1 to t

Each shock has a weight of 1 forever.

Mean of random walk with drift

E[Y(t)] = Y(0) + δt

With δ = 0 the mean is constant at Y(0), but the variance still grows.

Variance of random walk

Var(Y(t)) = tσ²

Assumes fixed Y(0). Standard deviation grows with √t.

Forecast of random walk

E[Y(t+h) | Y(t)] = Y(t) + hδ

Forecast variance of the error is hσ². For the plain walk, δ = 0.

First difference

ΔY(t) = Y(t) - Y(t-1) = δ + ε(t)

Stationary. Differencing removes the unit root.

Trend stationary process

Y(t) = a + bt + ε(t)

Shocks are temporary. Variance is constant at σ².

DF regression (no constant)

ΔY(t) = γY(t−1) + ε(t), with γ = φ − 1

Unit root means γ = 0.

DF regression with constant and trend

ΔY(t) = α + βt + γY(t−1) + ε(t)

Use when the series may have drift or a deterministic trend.

ADF regression

ΔY(t) = α + βt + γY(t−1) + Σ δ(i)ΔY(t−i) + ε(t)

Lagged differences remove serial correlation in errors.

Hypotheses

H0: γ = 0 (unit root); H1: γ < 0 (stationary)

One-sided test on the lower tail.

Test statistic

DF = γ̂ ÷ SE(γ̂)

Compare with Dickey-Fuller critical values, not standard t values.

Decision rule

Reject H0 if DF < critical value (more negative)

Rejection supports stationarity.

Random walk (with drift)

Yt = δ + Yt-1 + εt

Has a unit root. Variance of Yt grows with t. δ = 0 gives a pure random walk.

First difference

ΔYt = Yt - Yt-1

For a random walk with drift, ΔYt = δ + εt, which is stationary. Lose one observation.

Log return

rt = ln(Pt) - ln(Pt-1) = ln(Pt ÷ Pt-1)

First difference of log price. Usually stationary even when the price is not.

Deterministic trend model

Yt = a + b·t + εt

Trend-stationary. Remove the trend by detrending (use the residuals), not by differencing.

Spurious regression signature

High R² + significant t-statistics + very low Durbin-Watson

A warning sign only. Confirm with a unit root test such as Dickey-Fuller.

Second difference

Δ²Yt = ΔYt - ΔYt-1

Used only if the first difference is still nonstationary (series integrated of order 2).

Seasonal dummy regression (quarterly, with intercept)

Y(t) = β0 + γ2·D2(t) + γ3·D3(t) + γ4·D4(t) + ε(t)

Q1 is the base season. D(i) = 1 in quarter i, else 0. β0 is the Q1 mean; γ(i) is the Q(i) minus Q1 difference.

Number of dummies

Dummies needed = s − 1 (with intercept)

s is the number of seasons per year. Use s dummies only if you drop the intercept.

Seasonal means without intercept

Y(t) = α1·D1(t) + α2·D2(t) + α3·D3(t) + α4·D4(t) + ε(t)

Each α(i) is the mean of season i. Here α(i) = β0 + γ(i) from the intercept form.

Seasonal difference

ΔsY(t) = Y(t) − Y(t−s)

s = 4 for quarterly and 12 for monthly data. You lose s observations.

Seasonal AR term (idea)

Y(t) = φ·Y(t−s) + ε(t)

A significant autocorrelation at lag s, 2s and so on points to seasonality.

Chow test F-statistic

F = [(SSR_pooled − (SSR_1 + SSR_2)) ÷ k] ÷ [(SSR_1 + SSR_2) ÷ (n_1 + n_2 − 2k)]

k = number of estimated parameters including the intercept. Under no break, F follows F(k, n_1 + n_2 − 2k). Reject if F exceeds the critical value.

Break dummy regression

Y_t = β_0 + β_1·X_t + δ_0·D_t + δ_1·(D_t·X_t) + ε_t, with D_t = 0 before the break and 1 after

δ_0 shifts the intercept and δ_1 shifts the slope. Testing δ_0 = δ_1 = 0 jointly is equivalent to a Chow test.

Covariance stationarity conditions

E(Y_t) = μ; Var(Y_t) = σ²; Cov(Y_t, Y_t−h) = γ(h) for all t

A mean shift breaks the first condition. A change in the unconditional variance, such as a variance regime shift, breaks the second. Conditional variance that moves around a constant long-run level, as in GARCH with α + β < 1, does not.

GARCH(1,1) variance

σ²_t = ω + α·r²_t−1 + β·σ²_t−1

Stationary variance requires α + β < 1. Long-run variance = ω ÷ (1 − α − β).

EWMA variance

σ²_t = λ·σ²_t−1 + (1 − λ)·r²_t−1

A special case of GARCH with ω = 0 and α + β = 1. Variance has no mean reversion.

Quick revision

• Covariance stationarity means constant mean, constant variance and autocovariance that depends only on the lag.
• A deterministic trend is a predictable function of time; a trend-stationary series is stationary after removing it.
• A random walk, yₜ = yₜ₋₁ + εₜ, has a unit root, and its variance grows with time.
• With a unit root, shocks have permanent effects; with a stationary AR(1), shocks decay.
• Differencing a unit-root series once often makes it stationary.
• Dickey-Fuller null hypothesis: a unit root is present. Alternative: the series is stationary.
• Dickey-Fuller test statistics are compared with special critical values, not the normal t-table.
• Failing to reject the null does not prove a unit root; it only means you lack evidence against it.
• Spurious regression: unrelated nonstationary series can show a high R² and a significant slope.
• Seasonality is a regular pattern at a fixed calendar frequency, handled with seasonal dummies or seasonal adjustment.
• A structural break is a shift in parameters; estimates across the break can be misleading.
• Volatility often changes over time, so a constant-variance assumption can fail for returns.

Common mistakes

• Assuming a series with a constant mean is automatically stationary. Fix: Check all three: mean, variance and autocovariance.
• Thinking stationarity requires a normal distribution or zero mean. Fix: Covariance stationarity needs only a constant mean (any value), constant finite variance and lag-only autocovariance.
• Forgetting to exponentiate a log-linear forecast Fix: If the left-hand side is ln y, press e^x before reporting a level. Ask whether the answer is a log or a level.
• Reading β1 in a log-linear model as a change in units Fix: In ln y = β0 + β1t, β1 is the proportional growth per period. 0.012 means about 1.2% per period, not 0.012 units.
• Saying a random walk has infinite variance at any fixed time. Fix: At time t the variance is finite: tσ². It grows without limit as t increases, so no constant unconditional variance exists.
• Treating drift as a source of changing variance. Fix: Drift affects the mean: Y(0) + δt. Variance is tσ² with or without drift.
• Using standard t critical values such as −1.96 or −1.645. Fix: Under the unit root null the distribution is nonstandard. Always use the Dickey-Fuller critical values given in the question.
• Reading failure to reject as proof of a unit root. Fix: Say the data are consistent with a unit root. The test has low power, especially in small samples or when φ is near 1.
• Trusting a high R² and large t-statistic from a regression of two trending price series. Fix: Check for unit roots first. If both series have them, the t-statistics are unreliable. Difference before regressing.
• Detrending a random walk and calling it stationary. Fix: Detrending only removes a deterministic trend. A random walk has a stochastic trend, so you must difference it.

Exam tips

• Expect conceptual questions that ask which condition is violated. Name the specific condition.
• For AR(1), remember |φ| < 1 for stationarity and φ = 1 for a random walk.
• Link the topic to remedies: differencing for unit roots, detrending for deterministic trends, seasonal adjustment for seasonality.
• Watch for the spurious regression link: two trending series can show a high R-squared with no real relationship.
• Do not confuse covariance stationarity with strict stationarity. FRM uses the covariance form.
• Look at the left-hand side of the regression first. ln y versus y decides how you read every coefficient and whether you must exponentiate.
• Expect conceptual questions as well as arithmetic. Typical ones ask which trend fits constant percentage growth (log-linear) and which captures a changing slope (quadratic).
• Know the link to stationarity. A trend-stationary series becomes stationary after removing a deterministic trend. A unit-root series does not, and it needs differencing.