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FRM Exam Part I · Stationary Time Series

White Noise Process in Time Series Explained

Updated 11 October 2026 · Fact-checked

A white noise process is a sequence of random variables with zero mean, constant finite variance σ², and zero autocorrelation at every nonzero lag. Independent white noise adds full independence. Gaussian white noise adds normality, which makes uncorrelated mean independent. To solve questions, check the three conditions, then apply them.

Understand White Noise Processes

A white noise process is the simplest covariance stationary time series. It has no pattern you can predict from its past. It is the random shock that drives AR, MA and ARMA models.

A series ε_t is white noise if three conditions hold: E(ε_t) = 0 for all t, Var(ε_t) = σ² (constant and finite), and Cov(ε_t, ε_t-k) = 0 for all k ≠ 0. Because the variance is constant and the autocovariances do not depend on time, white noise is covariance stationary.

There are stronger versions. Independent white noise requires the ε_t to be independent, not just uncorrelated. Independence rules out any dependence, including nonlinear dependence such as volatility clustering in ε_t². Gaussian (normal) white noise is independent white noise where each ε_t is also normally distributed, written ε_t ~ N(0, σ²). For jointly normal variables, uncorrelated implies independent, so Gaussian white noise is automatically independent.

The order of strength is: Gaussian white noise ⇒ independent white noise ⇒ white noise. The reverse does not hold. A series can be uncorrelated yet dependent, for example when squared values are correlated. Such a series is white noise but not independent white noise.

White noise has a flat, zero autocorrelation function (ACF) and partial autocorrelation function (PACF) beyond lag 0. Its best forecast at any horizon is the mean, zero. Analysts check model residuals against white noise: if residuals are not white noise, the model has missed structure. The Ljung-Box and Box-Pierce Q-statistics test the joint hypothesis that several autocorrelations are zero.

Key formulas to remember

White noise conditions
E(ε_t) = 0; Var(ε_t) = σ²; Cov(ε_t, ε_t-k) = 0 for k ≠ 0
All three must hold. σ² must be finite and the same for every t.
Autocorrelation of white noise
ρ(0) = 1; ρ(k) = 0 for k ≥ 1
Lag 0 is always 1. All other lags are zero in the population.
Gaussian white noise
ε_t ~ i.i.d. N(0, σ²)
Normal plus uncorrelated gives independence.
Variance of a sum of white noise terms
Var(ε_1 + ε_2 + … + ε_n) = nσ²
Holds because covariances are zero. Standard deviation is σ√n.
Variance of a linear combination
Var(aε_t + bε_t-1) = (a² + b²)σ²
Cross term vanishes because Cov(ε_t, ε_t-1) = 0.
Approximate sample ACF band
±1.96 ÷ √T
For a white noise series of length T, about 95% of sample autocorrelations should fall inside this band.
Ljung-Box statistic
Q = T(T + 2) Σ [ρ̂(k)² ÷ (T − k)], k = 1 to m
Under the null of white noise, Q is approximately chi-squared with m degrees of freedom when you test a raw series. When you test the residuals of a fitted ARMA(p,q) model, the degrees of freedom are m − p − q, because the estimated parameters use up degrees of freedom.

How to solve White Noise Processes questions

Use this method for any question on white noise properties, classification or calculations.

  1. 1Identify what is asked: a property, a classification (white noise, independent, Gaussian) or a variance or probability calculation.
  2. 2Check the three core conditions: zero mean, constant finite variance, zero autocovariance at all nonzero lags.
  3. 3Decide how strong the assumption is. Only uncorrelated means white noise. Independent means independent white noise. Normal and independent means Gaussian white noise.
  4. 4For variances of sums or combinations, drop all covariance terms and add squared coefficients times σ².
  5. 5For probabilities, use the normal distribution only if the series is Gaussian. Standardize with z = x ÷ σ.
  6. 6For diagnostics, compare sample autocorrelations with ±1.96 ÷ √T or use the Ljung-Box Q against a chi-squared critical value.
  7. 7State the conclusion in words: for example, the residuals look like white noise, so the model captures the dynamics.

Quickest way: Three-check shortcut

When to use it: Use when the question asks you to classify a series or pick a true statement among four options.

  1. Ask: is the mean zero, the variance constant, and the correlation zero at every lag? If any fails, it is not white noise.
  2. Ask: does the statement claim independence? Then it needs more than uncorrelated, unless the series is Gaussian.
  3. For variance of a sum, multiply the count by σ² and skip covariances.
  4. For the ACF band, compute 1.96 ÷ √T and compare.

Common mistakes in White Noise Processes

  • Treating white noise as independent in all cases

    Uncorrelated and independent sound the same.

    Fix: Remember that independence is stronger. Only for normal variables does uncorrelated imply independent.

  • Assuming white noise must be normal

    Gaussian white noise is the common textbook case.

    Fix: White noise needs only zero mean, constant variance and no autocorrelation. Any distribution with those moments qualifies.

  • Forgetting that ρ(0) = 1

    Students say all autocorrelations are zero.

    Fix: Zero applies to lags k ≥ 1. Lag 0 is the correlation of the series with itself.

  • Adding covariance terms when finding the variance of a sum

    Habit from portfolio variance formulas.

    Fix: For white noise all covariances are zero, so only variances remain.

  • Allowing a nonzero mean or time-varying variance

    Confusing white noise with any random series.

    Fix: A series with a nonzero mean or a changing unconditional variance is not white noise. Volatility clustering in a series with constant unconditional variance and no autocorrelation is still white noise, but it is not independent white noise.

  • Saying a random walk is white noise

    Both come from random shocks.

    Fix: A random walk is a cumulative sum of white noise. It is nonstationary. Its first difference is white noise.

Worked examples

Example 1

ε_t is Gaussian white noise with σ² = 4. Define Y_t = 3ε_t − 2ε_t-1. Find Var(Y_t) and the probability that Y_t exceeds 6.5 (use N(0.90) ≈ 0.8159 after standardizing; Y_t has mean 0).

Show the solution
  1. Variance of a linear combination: Var(Y_t) = (3² + (−2)²)σ².
  2. Cross term is zero because Cov(ε_t, ε_t-1) = 0.
  3. Var(Y_t) = (9 + 4) × 4 = 52.
  4. Standard deviation = √52 ≈ 7.211.
  5. Y_t is a linear combination of normal variables, so it is normal with mean 0.
  6. z = 6.5 ÷ 7.211 ≈ 0.90, so P(Y_t > 6.5) = 1 − N(0.90) ≈ 1 − 0.8159 = 0.1841.

Answer: Var(Y_t) = 52. P(Y_t > 6.5) ≈ 0.184 (z ≈ 0.90).

Example 2

A model's residual series has T = 100 observations. Which sample autocorrelation, in absolute value, would be outside the approximate 95% band for white noise: 0.15, 0.18, 0.21 or 0.05?

Show the solution
  1. Band = ±1.96 ÷ √T.
  2. √100 = 10, so band = ±1.96 ÷ 10 = ±0.196.
  3. Compare each value: 0.05, 0.15 and 0.18 are inside 0.196.
  4. 0.21 is greater than 0.196, so it lies outside.

Answer: 0.21 is outside the band, so it suggests significant autocorrelation at that lag.

Exam tips

  • Know the hierarchy: Gaussian white noise ⇒ independent white noise ⇒ white noise. Questions often test which implication fails.
  • Expect variance-of-a-combination questions. Use squared coefficients times σ² and no covariance terms.
  • Remember that white noise is the building block: AR, MA and ARMA models are built from it, and good residuals should look like it.
  • For diagnostics, memorize ±1.96 ÷ √T and know the Ljung-Box null is that autocorrelations up to lag m are all zero.
  • Read wording closely: 'uncorrelated' versus 'independent' decides the correct option.

Practice questions from Stationary Time Series

White Noise Processes: frequently asked questions

What is white noise in time series?

It is a series with zero mean, constant finite variance and no autocorrelation at any nonzero lag. It has no predictable pattern. It is the random shock term in models like AR and MA.

What is the difference between white noise and independent white noise?

White noise requires only that the terms are uncorrelated. Independent white noise requires full independence, which also rules out nonlinear dependence such as correlated squared values. Independence implies uncorrelated, but not the other way round in general.

Is Gaussian white noise the same as independent white noise?

Gaussian white noise is a special case of independent white noise in which each term is also normal. For jointly normal variables, uncorrelated implies independent, so the two properties coincide there.

Is white noise stationary?

Yes. Its mean and variance are constant and its autocovariances depend only on the lag, so it is covariance stationary.

How do I test whether residuals are white noise?

Inspect the sample ACF against the ±1.96 ÷ √T band and use a Ljung-Box or Box-Pierce Q test. If you fail to reject the null of zero autocorrelations, the residuals are consistent with white noise.