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FRM Exam Part I · Nonstationary Time Series

Stationary vs Nonstationary 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 finite and constant, and its autocovariance depends only on the lag, not on time. Nonstationary series break one or more of these through trends, changing variance or structural breaks. Check each condition in turn to classify a series.

Understand Stationary vs Nonstationary Time Series

A time series is a set of observations ordered in time, such as daily returns or monthly GDP. To model it, you need its behaviour to be stable enough that the past tells you something about the future. That stability is called stationarity.

FRM focuses on covariance stationarity (weak stationarity). A series is covariance stationary if three things hold:

  • The mean is constant over time.
  • The variance is finite and constant over time.
  • The covariance between Y(t) and Y(t-k) depends only on the lag k, not on t.

Think of a series that keeps returning to the same average level with the same spread. Daily equity returns are often close to this. A stock price level is not: it drifts and wanders, so its mean and variance change over time.

A series is nonstationary if it breaks any condition. Common causes are: a deterministic trend (mean rises with time, e.g. Y(t) = a + bt + e(t)); a random walk with a unit root (shocks never fade and variance grows with time); seasonality (the mean differs by season); changing variance (volatility clusters or shifts); and structural breaks (the mean or parameters change at some date, such as after a regime change or crisis).

Why it matters: estimates of means, autocorrelations and regression coefficients rely on stable relationships. With nonstationary data, sample statistics do not converge to a meaningful value, forecasts can be unreliable, and regressions of one trending series on another can show a high R-squared and significant t-statistics with no real link (spurious regression). Typical fixes are detrending, differencing, seasonal adjustment or modelling breaks.

Note that covariance stationarity is about the first two moments only. It does not require a normal distribution, and it does not mean the series is unpredictable.

Key formulas to remember

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.

How to solve Stationary vs Nonstationary Time Series questions

Use this checklist for any question asking whether a series is stationary or what to do about it.

  1. 1Write down the process or read the description of the data (formula, chart, or statistics by sub-period).
  2. 2Compute or identify the mean. Does it depend on t (trend, seasonal pattern, level shift)? If yes, it is nonstationary.
  3. 3Compute or identify the variance. Is it finite and the same across time? A growing variance (random walk) or shifting volatility fails.
  4. 4Check the autocovariance. Does it depend only on the lag? If it depends on t, fail.
  5. 5Look for a unit root or a break: a coefficient of 1 on the lagged value, or a date where parameters change.
  6. 6State the conclusion: stationary only if all three conditions hold.
  7. 7If asked for a remedy, match it to the cause: detrend for deterministic trend, difference for unit root, seasonal adjustment for seasonality, split or dummy for structural breaks.

Quickest way: Three-condition scan

When to use it: Multiple-choice questions with a short process or a description of a series.

  1. Ask: does the mean move with time? Trend, seasonal or level shift means nonstationary.
  2. Ask: does the variance grow or shift? A coefficient of 1 on Y(t-1) means a random walk, so nonstationary.
  3. For AR(1) Y(t) = c + φY(t-1) + e(t), the series is stationary if |φ| < 1, and then the mean is c ÷ (1 - φ).
  4. Eliminate options that call a series stationary just because it is bounded, normal or has zero mean.

Common mistakes in Stationary vs Nonstationary Time Series

  • Assuming a series with a constant mean is automatically stationary.

    Students remember only the mean condition.

    Fix: Check all three: mean, variance and autocovariance.

  • Thinking stationarity requires a normal distribution or zero mean.

    Confusing covariance stationarity with strict stationarity or with normality assumptions.

    Fix: Covariance stationarity needs only a constant mean (any value), constant finite variance and lag-only autocovariance.

  • Treating a random walk with drift as stationary because it has a fixed drift.

    A constant drift looks like a stable feature.

    Fix: A unit root makes the variance grow with time and the mean trend, so it is nonstationary. Difference it.

  • Applying a deterministic detrend to a unit-root series.

    Both show upward drift on a chart.

    Fix: Detrending fixes a trend-stationary series. Differencing fixes a unit root.

  • Ignoring structural breaks when the full-sample statistics look fine.

    Averages over the whole sample hide a shift at a date.

    Fix: Compare sub-period means and variances. A change signals a break, which violates constant mean or variance.

Worked examples

Example 1

Y(t) = 2 + 0.6·Y(t-1) + e(t), where e(t) is white noise with variance 4. Is Y covariance stationary? If so, find its mean and variance.

Show the solution
  1. The AR(1) coefficient is φ = 0.6, and |φ| < 1, so the series is covariance stationary.
  2. Mean: μ = c ÷ (1 - φ) = 2 ÷ (1 - 0.6) = 2 ÷ 0.4 = 5.
  3. Variance: σ²÷(1 - φ²) = 4 ÷ (1 - 0.36) = 4 ÷ 0.64 = 6.25.

Answer: Yes. Mean = 5 and variance = 6.25.

Example 2

A random walk is Y(t) = Y(t-1) + e(t), Y(0) = 0, with Var[e(t)] = 0.25. Find Var[Y(16)] and state whether the series is covariance stationary.

Show the solution
  1. Y(16) = e(1) + e(2) + ... + e(16), a sum of 16 independent shocks.
  2. Var[Y(16)] = 16 × 0.25 = 4.
  3. Var[Y(t)] = 0.25t, which depends on t, so the variance is not constant.

Answer: Var[Y(16)] = 4. The series is not covariance stationary because its variance grows with time.

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.

Practice questions from Nonstationary Time Series

Stationary vs Nonstationary Time Series: frequently asked questions

What is the difference between stationary and nonstationary time series?

A stationary series has a constant mean, constant finite variance and autocovariance that depends only on the lag. A nonstationary series breaks at least one of these, for example through a trend, a unit root, changing volatility or a structural break.

Why is stationarity important in time series modelling?

Models such as AR and ARMA estimate stable relationships from past data. If the mean, variance or autocovariances change over time, estimates and forecasts become unreliable, and regressions can be spurious.

Is a random walk covariance stationary?

No. A random walk has a unit root, so shocks persist and its variance grows with time. First differencing a random walk gives a stationary series.

Does a stationary series have to be normally distributed?

No. Covariance stationarity restricts only the mean, variance and autocovariances. The distribution can take any shape.