FRM Exam Part I · Stationary Time Series

Covariance Stationary Time Series for FRM Part I

Updated 11 October 2026 · Fact-checked

A time series is covariance stationary if its mean is constant, its variance is constant and finite, and its autocovariance depends only on the lag between observations, not on time. You check these three conditions, then use the ACF and PACF to pick a model such as AR, MA or ARMA.

Understand Covariance Stationary Time Series

A time series is a sequence of observations ordered in time, such as daily returns or monthly inflation. To model it, you need the series to behave the same way in the past and the future. Without that, estimates from past data tell you nothing reliable about tomorrow.

Covariance stationarity is the weak form of that stability. It requires three things: a constant mean, a constant and finite variance, and an autocovariance that depends only on the lag (the gap between two points) and not on the date. It is called weak because it only restricts the first two moments, not the full distribution.

Why it matters: AR, MA and ARMA models are built for covariance stationary data. If the series trends or its variance drifts, the sample mean and sample autocorrelations are misleading, and forecasts do not settle on a long-run level. Stock prices usually fail the test. Returns usually pass it more closely.

To describe the dependence, you use the autocovariance function γ(k) = Cov(Yt, Yt-k) and the autocorrelation function (ACF) ρ(k) = γ(k) ÷ γ(0). The partial autocorrelation function (PACF) measures the correlation between Yt and Yt-k after removing the effect of the shorter lags in between. For a stationary series, ρ(0) = 1 and the ACF decays toward zero as the lag grows.

A white noise process is the simplest covariance stationary series: mean zero, constant variance σ², and zero autocorrelation at every non-zero lag. More complex stationary models are built from white noise shocks.

Key formulas to remember

Constant mean: E(Yt) = μ for all t; The expected value does not depend on time.
Constant, finite variance: Var(Yt) = γ(0) = σ² < ∞ for all t; Variance is the autocovariance at lag 0.
Autocovariance depends only on lag: γ(k) = Cov(Yt, Yt-k), the same for every t; Depends on k, not on the date t. Also γ(k) = γ(-k).
Autocorrelation function: ρ(k) = γ(k) ÷ γ(0); Lies between -1 and +1, and ρ(0) = 1.
White noise: E(εt) = 0; Var(εt) = σ²; Cov(εt, εt-k) = 0 for k ≠ 0; A basic covariance stationary process.
AR(1) stationarity condition: Yt = c + φYt-1 + εt is stationary if |φ| < 1; Then the mean is c ÷ (1 - φ) and the ACF is ρ(k) = φ^k.
AR(1) variance: Var(Yt) = σ² ÷ (1 - φ²); Valid only when |φ| < 1.
Approximate ACF significance band: ± 1.96 ÷ √T; Rough 95% band for a sample autocorrelation under white noise, with T observations.

How to solve Covariance Stationary Time Series questions

Use this sequence for any question that asks whether a series is covariance stationary or how its ACF and PACF behave.

1Write down the process equation and identify the mean, the variance and any time-dependent terms.
2Check the mean: does E(Yt) depend on t? A deterministic trend or a changing intercept means it is not stationary.
3Check the variance: is it constant and finite? A random walk has variance that grows with t, so it fails.
4Check the autocovariance: does Cov(Yt, Yt-k) depend only on k? If it depends on t, the series fails.
5For AR models, test the coefficient condition (for AR(1), |φ| < 1). A root at 1 is a unit root and means non-stationary.
6If the question asks about the ACF or PACF, use the pattern: AR(p) has a decaying ACF and PACF that cuts off after lag p; MA(q) has an ACF that cuts off after lag q and a decaying PACF.
7State the conclusion and its consequence, such as differencing a non-stationary series before modelling.

Quickest way: Three-condition scan

When to use it: Use this when the question gives a short process equation and asks which series is stationary.

Look for a time trend or a drift term that grows with t. If present, reject.
Look for a coefficient of 1 on the lagged value (Yt = Yt-1 + εt). That is a random walk, so reject.
For AR(1), accept if |φ| < 1. Any MA with finite coefficients is stationary.
For ACF/PACF patterns, remember: AR cuts off in PACF, MA cuts off in ACF.

Common mistakes in Covariance Stationary Time Series

Saying a stationary series has constant values or no randomness.
The word stationary sounds like standing still.
Fix: Stationary means the statistical properties (mean, variance, autocovariance) are stable. The series can still fluctuate a lot around its mean.
Forgetting that the variance must be finite as well as constant.
Students memorise 'constant mean and variance' only.
Fix: Include 'finite' in your definition. It matters for heavy-tailed series.
Treating |φ| = 1 as stationary for an AR(1).
The boundary case looks close enough.
Fix: The condition is strict: |φ| < 1. At φ = 1 you have a random walk with a unit root, which is not covariance stationary.
Mixing up the ACF and PACF cut-off patterns for AR and MA models.
Both patterns use the words cut off and decay.
Fix: Remember: AR(p) PACF cuts off after lag p. MA(q) ACF cuts off after lag q.
Assuming asset prices are covariance stationary because returns are.
Prices and returns are used interchangeably in conversation.
Fix: Prices typically trend and behave like random walks. Returns (the differences) are often close to stationary.
Using γ(k) instead of ρ(k) when a question asks for the correlation.
The two notations look similar.
Fix: Divide by γ(0): ρ(k) = γ(k) ÷ γ(0). Correlation is unitless and lies in [-1, 1].

Worked examples

Example 1

An AR(1) process is Yt = 2 + 0.6Yt-1 + εt, where εt is white noise with variance 4. Is it covariance stationary? If so, find its mean, variance and the autocorrelation at lag 2.

Show the solution

Check the condition: |φ| = 0.6 < 1, so the process is covariance stationary.
Mean: μ = c ÷ (1 - φ) = 2 ÷ (1 - 0.6) = 2 ÷ 0.4 = 5.
Variance: σ² ÷ (1 - φ²) = 4 ÷ (1 - 0.36) = 4 ÷ 0.64 = 6.25.
Autocorrelation at lag 2: ρ(2) = φ² = 0.36.

Answer: Stationary; mean = 5, variance = 6.25, ρ(2) = 0.36.

Example 2

Which of the following is covariance stationary? (A) Yt = 0.5t + εt (B) Yt = Yt-1 + εt (C) Yt = 3 + εt + 0.4εt-1 (D) Yt = 1 + 1.2Yt-1 + εt, where εt is white noise in each case.

Show the solution

(A) The mean is 0.5t, which changes with time. Not stationary.
(B) This is a random walk with φ = 1. Variance grows with t. Not stationary.
(C) This is an MA(1) with a finite coefficient. Mean = 3, variance = σ²(1 + 0.16) = 1.16σ², and autocovariance depends only on lag. Stationary.
(D) φ = 1.2 and |φ| > 1, so the process is explosive. Not stationary.

Answer: (C) is covariance stationary.

Exam tips

Expect questions that give a process equation and ask which option is stationary. Scan for trends and a coefficient of 1 or more first.
Memorise the AR and MA patterns: PACF cut-off signals AR, ACF cut-off signals MA.
For AR(1) questions, compute the mean as c ÷ (1 - φ) and the variance as σ² ÷ (1 - φ²) before looking at the options.
Read carefully whether the question asks for autocovariance or autocorrelation, and divide by γ(0) when needed.
When a series is not stationary, the usual remedy in answers is differencing or detrending, so be ready to name it.

Practice questions from Stationary Time Series

Covariance Stationary Time Series: frequently asked questions

What are the conditions for covariance stationarity?

The mean must be constant over time, the variance must be constant and finite, and the autocovariance between two observations must depend only on the lag between them. If any one fails, the series is not covariance stationary.

What is the difference between stationary and non-stationary time series?

A stationary series has stable mean, variance and autocovariance, so it tends to revert to a long-run level. A non-stationary series, such as a random walk or a trending series, has moments that change over time, and past patterns do not carry over reliably.

What is the difference between the ACF and the PACF?

The ACF shows the correlation between Yt and Yt-k, including indirect effects through the lags in between. The PACF shows only the direct correlation at lag k after removing those intermediate lags. Together they help you choose between AR and MA models.

Why does covariance stationarity matter for forecasting?

Models such as AR, MA and ARMA rely on stable relationships over time. If the series is stationary, sample estimates are meaningful and forecasts converge to the long-run mean. If it is not, estimates can be misleading and may produce spurious results.