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FRM Exam Part I · Nonstationary Time Series

Random Walks and Unit Roots for FRM Part I

Updated 11 October 2026 · Fact-checked

A random walk is a series where today's value equals yesterday's value plus a white noise shock: Y(t) = Y(t-1) + ε(t). It has a unit root (coefficient of 1), so shocks never fade and variance grows with time (tσ²). Add a constant to get drift. Differencing the series makes it stationary.

Understand Random Walks and Unit Roots

Start with an AR(1) model: Y(t) = δ + φY(t-1) + ε(t). If |φ| < 1, shocks fade and the series returns to its mean. It is covariance stationary. If φ = 1, the series has a unit root. That is a random walk.

A random walk is Y(t) = Y(t-1) + ε(t), where ε is white noise with variance σ². Each period's value is the last value plus a new random step. A random walk with drift adds a constant: Y(t) = δ + Y(t-1) + ε(t). The drift δ is the average step. Each period the series moves up by δ on average (or down if δ is negative), plus noise.

Why are shocks permanent? Substitute backward. Y(t) = Y(0) + ε(1) + ε(2) + ... + ε(t) for the plain walk. Every past shock enters with a weight of 1. Nothing decays. A shock today shifts the best forecast of every future value by the same amount. In a stationary AR(1), the same shock would have weight φ^k after k periods and would vanish.

Why does variance grow? The shocks are independent, so Var(Y(t)) = tσ², given a fixed starting value Y(0). It grows linearly with time. It does not converge to a finite number as t grows without limit, so there is no constant unconditional variance. The mean also depends on t when drift is present: E[Y(t)] = Y(0) + δt. A series with no constant mean or variance is nonstationary.

Do not confuse this with a trend stationary process: Y(t) = a + bt + ε(t). It has a deterministic trend, but shocks are temporary. Deviations from the trend line die out, and variance is constant at σ². Detrending (subtracting a + bt) makes it stationary. A random walk with drift needs differencing instead. The change ΔY(t) = δ + ε(t) is stationary. Both can look like upward-trending lines on a chart, but their shock behaviour and the cure differ.

Key formulas to remember

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.

How to solve Random Walks and Unit Roots questions

Use this method for any question on random walks, drift or unit roots.

  1. 1Write the process as Y(t) = δ + φY(t-1) + ε(t) and identify φ and δ.
  2. 2If φ = 1, it has a unit root. If |φ| < 1, it is stationary around its mean. Then check whether a deterministic trend term bt appears.
  3. 3If asked about shocks, use backward substitution. A unit root means weight 1 on every past shock, so effects are permanent. A stationary AR(1) has weight φ^k, so effects decay.
  4. 4For the mean, compute Y(0) + δt. For variance, compute tσ². For a forecast h steps ahead, use the last value plus hδ, with error variance hσ².
  5. 5For the standard deviation over a horizon, take σ√t or σ√h, not σ times t.
  6. 6To fix nonstationarity, difference a unit root series. Detrend a trend stationary series. Pick the remedy that matches the process.
  7. 7Check that your answer is consistent: variance rising with time means nonstationary, constant variance means stationary.

Quickest way: Coefficient and shock-weight shortcut

When to use it: Use it for conceptual MCQs asking whether a series is stationary, whether shocks persist, or how variance behaves.

  1. Look at the coefficient on Y(t-1). Exactly 1 means unit root and nonstationary.
  2. Is there a bt term with no unit root? Then it is trend stationary: temporary shocks, constant variance.
  3. Variance of a random walk after t periods is tσ². Standard deviation is σ√t. Double the horizon and the standard deviation rises by √2.
  4. Forecast = last value + hδ. For plain random walk, the forecast is the last value.
  5. Remedy: difference for unit root, detrend for deterministic trend.

Common mistakes in Random Walks and Unit Roots

  • Saying a random walk has infinite variance at any fixed time.

    The phrase 'variance grows without bound' gets shortened to 'infinite variance'.

    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.

    Drift makes the series trend, so students assume it also changes the spread.

    Fix: Drift affects the mean: Y(0) + δt. Variance is tσ² with or without drift.

  • Confusing a random walk with drift with a trend stationary process.

    Both charts trend upward.

    Fix: Ask what happens to a shock. In a random walk it is permanent and variance grows. In a trend stationary process it fades and variance is constant. Difference the first, detrend the second.

  • Scaling the standard deviation by t instead of √t.

    Students mix up variance (proportional to t) with standard deviation.

    Fix: Variance = tσ². Standard deviation = σ√t. Take the square root at the end.

  • Thinking a unit root means φ is merely close to 1, or that |φ| = 1 only matters if φ is positive.

    Near-unit-root series look very persistent, so the distinction blurs.

    Fix: A unit root means φ = 1 exactly. φ = 0.98 is stationary in theory, though it behaves like a random walk in small samples and is hard to tell apart in tests.

  • Forecasting a random walk by reverting to a long-run mean.

    Stationary AR models revert to a mean, and the habit carries over.

    Fix: A random walk has no mean to revert to. The forecast is the current value plus hδ.

Worked examples

Example 1

A series follows Y(t) = 0.5 + Y(t-1) + ε(t), where ε is white noise with σ² = 4. Y(0) = 100. Find the expected value and the standard deviation of Y(25).

Show the solution
  1. The coefficient on Y(t-1) is 1, so this is a random walk with drift δ = 0.5.
  2. Mean: E[Y(25)] = Y(0) + δt = 100 + 0.5 × 25 = 112.5.
  3. Variance: Var(Y(25)) = tσ² = 25 × 4 = 100.
  4. Standard deviation = √100 = 10.

Answer: E[Y(25)] = 112.5 and the standard deviation is 10.

Example 2

Y(t) = Y(t-1) + ε(t) with σ = 2% per period. Y(t) is 80 today. Which statement is correct about the forecast and its uncertainty four periods ahead?
A. Forecast 80, forecast error standard deviation 4%
B. Forecast 80, forecast error standard deviation 8%
C. Forecast 80, forecast error standard deviation 16%
D. Forecast 76, forecast error standard deviation 4%

Show the solution
  1. The process is a random walk with no drift, so the forecast at any horizon is today's value: 80.
  2. The forecast error is the sum of four independent shocks. Its variance is hσ² = 4 × (2%)² = 4 × 4 = 16 (in %²).
  3. The standard deviation is √16 = 4%, which equals σ√h = 2% × 2.
  4. Option B uses σ × h = 8%, which is wrong. Option C uses variance 16 as a standard deviation. Option D wrongly shifts the forecast down.

Answer: A. Forecast 80, standard deviation 4%.

Exam tips

  • Questions often test the contrast: unit root means permanent shocks and growing variance, stationary AR(1) means decaying shocks and constant variance.
  • Expect to compute tσ² or hσ² and then take the square root when asked for a standard deviation. Do the square root last.
  • Read whether the process includes a bt term. That single term separates trend stationary from a random walk with drift.
  • Match remedy to process: difference for a unit root, detrend for a deterministic trend. A wrong remedy is a common distractor.
  • For a stationary AR(1), recall the mean is δ ÷ (1 - φ). A question may ask why that formula fails when φ = 1: the denominator is zero.

Practice questions from Nonstationary Time Series

Random Walks and Unit Roots in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Random Walks and Unit Roots: frequently asked questions

What is a unit root in simple terms?

It means the coefficient on the lagged value in an AR model equals exactly 1. Past shocks then carry full weight forever, so the series does not return to a fixed mean. The series is nonstationary.

Why does a random walk have growing variance?

The value is the starting point plus the sum of all past independent shocks. Variances of independent shocks add, so Var(Y(t)) = tσ². Each period adds another σ², so it keeps rising.

What is the difference between a random walk with drift and a trend stationary process?

Both can trend upward. In a random walk with drift, shocks are permanent and variance grows, so you difference the series. In a trend stationary process, shocks fade and variance is constant, so you remove the trend line.

Does drift make a random walk stationary or change its variance?

No. Drift only moves the mean: E[Y(t)] = Y(0) + δt. The variance is still tσ², and the series stays nonstationary.