Risk Modelling and Survival Analysis · Applications of time series models

Time Series Components and Stationarity Explained

Q: What is the difference between strict and weak stationarity?

Strict stationarity requires every joint distribution to be unchanged by a time shift. Weak stationarity only requires a constant mean and autocovariance that depends on lag alone. Strict implies weak when the variance is finite, but not the other way round, except for Gaussian processes.

Q: How do I test whether a time series is stationary?

Plot the data and look for trend, changing spread or seasonality. Then examine the sample ACF: a slow decay points to non-stationarity. For a model, check the mean and autocovariance directly. Differencing and re-checking is the usual remedy.

Q: What is the difference between ACF and PACF?

The ACF at lag k is the correlation between X_t and X_t+k. The PACF at lag k is the same correlation after removing the linear effect of the observations between them. They are used together to identify AR and MA models.

Q: Is a random walk stationary?

No. Its variance grows with time, since Var(X_t) = tσ², so it fails the constant variance condition. Its first difference is white noise, which is stationary.

Q: Can a series with seasonality be stationary?

Not if the seasonal pattern is a deterministic component that changes the mean over the cycle. Remove it by seasonal differencing or by estimating and subtracting the seasonal effect, then check the remainder.

Updated 11 October 2026 · Fact-checked

A time series splits into trend, seasonal and random components. A series is weakly stationary if its mean is constant and its autocovariance depends only on the lag. It is strictly stationary if all joint distributions are unchanged by time shifts. To check, test the mean, variance, autocovariance and the sample ACF.

Understand Time Series Components and Stationarity

A time series is a set of observations X_t recorded in time order, such as monthly claim counts or daily share prices. The order matters, because nearby values are usually related. That is why ordinary independent-sample methods do not work well here.

A series is often thought of as a sum of parts. The trend is the long-run movement, up or down. The seasonal component is a pattern that repeats over a fixed period, such as higher motor claims every monsoon. The random (irregular) component is what remains after trend and seasonality are removed. In an additive model, X_t = trend + seasonal + random. In a multiplicative model, the parts multiply, which suits series whose seasonal swings grow with the level.

To model the random part with ARMA-type models, we need it to be stationary: its statistical behaviour does not change over time. Strict stationarity means the joint distribution of (X_t1, ..., X_tk) is the same as that of (X_t1+h, ..., X_tk+h) for all k, all times and all shifts h. Weak (second-order) stationarity asks for less: the mean is constant, the variance is finite, and the covariance between X_t and X_t+k depends only on the lag k, not on t.

Strict stationarity with finite second moments implies weak stationarity. The reverse is not true in general. The exception is a Gaussian process, where weak stationarity does imply strict stationarity, because a normal distribution is fully set by its mean and covariances.

The autocovariance at lag k measures how X_t and X_t+k move together. The autocorrelation function (ACF) scales it by the variance, so it lies between -1 and 1 and equals 1 at lag 0. The partial autocorrelation function (PACF) at lag k is the correlation between X_t and X_t+k after removing the linear effect of the values in between. ACF and PACF are the main tools for spotting non-stationarity and for choosing a model.

Key rules to remember

Additive decomposition: X_t = m_t + s_t + Z_t; m_t is trend, s_t is seasonal (repeats every d periods, so s_t+d = s_t), Z_t is random.
Multiplicative decomposition: X_t = m_t × s_t × Z_t; Taking logs turns it into an additive model.
Weak stationarity conditions: E[X_t] = μ for all t; Var(X_t) < ∞; Cov(X_t, X_t+k) = γ_k depends only on k; All three must hold. Mean cannot depend on t.
Autocovariance function: γ_k = Cov(X_t, X_t+k) = E[(X_t − μ)(X_t+k − μ)]; γ_0 = Var(X_t). Also γ_−k = γ_k.
Autocorrelation function: ρ_k = γ_k ÷ γ_0; ρ_0 = 1 and |ρ_k| ≤ 1.
Sample autocovariance: c_k = (1/n) Σ (x_t − x̄)(x_t+k − x̄), sum over t = 1 to n − k; Divide by n, as in the standard sample estimator. Sample ACF is r_k = c_k ÷ c_0.
Differencing: ∇X_t = X_t − X_t−1; seasonal: ∇_d X_t = X_t − X_t−d; First differences remove a linear trend; seasonal differences remove a seasonal pattern of period d.

How to solve Time Series Components and Stationarity questions

Use this order for any question asking whether a series or process is stationary, or asking you to describe its components.

1Write down the definition you need to test: weak stationarity (constant mean, finite variance, autocovariance depends on lag only) unless the question says strict.
2Find E[X_t]. If it depends on t, stop: the process is not stationary.
3Find Var(X_t) and check it is finite and constant in t.
4Find Cov(X_t, X_t+k) and see whether it depends only on k. Expand carefully using the properties of covariance.
5State the conclusion clearly. If the process is Gaussian, you may extend weak stationarity to strict stationarity.
6For data, plot the series and inspect the sample ACF. Slow decay of the ACF or a visible trend suggests non-stationarity; a regular pattern at lag d suggests seasonality.
7If non-stationary, propose a transformation (differencing, seasonal differencing, or logs first if variance grows with the level) and say how you would recheck.

Quickest way: Mean first, then covariance

When to use it: Use in multiple-choice questions and short written parts where you must decide quickly if a process is stationary.

Check the mean. A term like a·t, or a random walk with drift, makes it fail at once.
Check whether the variance grows with t. A random walk has Var(X_t) = tσ², so it is not stationary.
If the process is a sum of independent terms, compute the covariance by matching common terms only.
For a deterministic seasonal or trend term, expect non-stationarity unless it is removed.
Remember: the answer depends on lag only, never on t.

Common mistakes in Time Series Components and Stationarity

Saying a process is stationary because the mean is constant.
The mean is the easiest condition to check, so students stop there.
Fix: Always check variance and autocovariance as well. A constant mean alone is not enough.
Claiming weak stationarity implies strict stationarity.
The two terms sound like the same idea with different strength.
Fix: Only strict (with finite variance) implies weak. The converse holds in general only for Gaussian processes.
Letting the autocovariance depend on t in the final answer.
Students leave the answer in terms of t and s instead of the difference.
Fix: Simplify to a function of k = difference in times. If t remains, the process is not weakly stationary.
Confusing ACF and PACF.
Both are plotted against lag and look similar.
Fix: ACF is the full correlation at each lag. PACF removes the effect of intermediate lags. Both equal the same value at lag 1.
Treating a decaying ACF in a sample plot as proof of stationarity, or a few small spikes as proof of non-stationarity.
Sample ACF values have random error.
Fix: Compare with the approximate ±2/√n limits for white noise, and look at the overall pattern, not single spikes.
Dividing by n − k when computing the sample autocovariance.
It feels natural to average over the number of terms in the sum.
Fix: Use the standard form with divisor n, as in the IAI notes, unless the question states otherwise.

Worked examples

Example 1

Let Z_t be independent N(0, 4) random variables. Define X_t = 3 + Z_t + 0.5 Z_t−1. Show that X_t is weakly stationary and find its autocorrelation at lags 1 and 2.

Show the solution

Mean: E[X_t] = 3 + 0 + 0 = 3, constant in t.
Variance: Var(X_t) = Var(Z_t) + 0.25 Var(Z_t−1) = 4 + 0.25 × 4 = 5. This is finite and constant.
Lag 1: Cov(X_t, X_t+1) = Cov(Z_t + 0.5 Z_t−1, Z_t+1 + 0.5 Z_t). The only common term is Z_t, with coefficients 1 (in the first) and 0.5 (in the second). So γ_1 = 0.5 × 4 = 2.
Lag 2: X_t involves Z_t and Z_t−1; X_t+2 involves Z_t+2 and Z_t+1. No common terms, so γ_2 = 0.
The autocovariances depend only on the lag, so X_t is weakly stationary. Also ρ_1 = 2 ÷ 5 = 0.4 and ρ_2 = 0.

Answer: X_t is weakly stationary with mean 3 and variance 5. ρ_1 = 0.4 and ρ_2 = 0.

Example 2

Let Z_t be independent with mean 0 and variance σ². Define X_t = a + bt + Z_t, where b ≠ 0. Is X_t weakly stationary? If not, find a transformation that gives a stationary series.

Show the solution

Mean: E[X_t] = a + bt. This depends on t because b ≠ 0.
A constant mean is required for weak stationarity, so X_t is not weakly stationary.
Take first differences: Y_t = X_t − X_t−1 = b + Z_t − Z_t−1.
E[Y_t] = b, constant. Var(Y_t) = σ² + σ² = 2σ², constant.
Lag 1: Cov(Y_t, Y_t+1) = Cov(Z_t − Z_t−1, Z_t+1 − Z_t) = −Var(Z_t) = −σ². For lags of 2 or more there are no common terms, so the covariance is 0.
These depend only on the lag, so Y_t is weakly stationary.

Answer: X_t is not weakly stationary because its mean a + bt depends on t. The first difference Y_t = b + Z_t − Z_t−1 is weakly stationary, with mean b, variance 2σ² and lag 1 autocorrelation −0.5.

Exam tips

Write the three conditions for weak stationarity explicitly and tick them off. Examiners award marks for each condition.
In covariance questions, list the terms of each expression and match only the common independent terms. This avoids long expansions.
When asked to comment on a sample ACF plot, mention slow decay, seasonal spikes at multiples of d, and the ±2/√n guide lines.
State assumptions, such as Gaussian errors, when you move from weak to strict stationarity.
In computer-based paper work, plot the series and ACF first and comment before fitting any model.

Practice questions from Applications of time series models

Time Series Components and Stationarity in other exams

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

Time Series Components and Stationarity: frequently asked questions

What is the difference between strict and weak stationarity?

Strict stationarity requires every joint distribution to be unchanged by a time shift. Weak stationarity only requires a constant mean and autocovariance that depends on lag alone. Strict implies weak when the variance is finite, but not the other way round, except for Gaussian processes.

How do I test whether a time series is stationary?

Plot the data and look for trend, changing spread or seasonality. Then examine the sample ACF: a slow decay points to non-stationarity. For a model, check the mean and autocovariance directly. Differencing and re-checking is the usual remedy.

What is the difference between ACF and PACF?

The ACF at lag k is the correlation between X_t and X_t+k. The PACF at lag k is the same correlation after removing the linear effect of the observations between them. They are used together to identify AR and MA models.

Is a random walk stationary?

No. Its variance grows with time, since Var(X_t) = tσ², so it fails the constant variance condition. Its first difference is white noise, which is stationary.

Can a series with seasonality be stationary?

Not if the seasonal pattern is a deterministic component that changes the mean over the cycle. Remove it by seasonal differencing or by estimating and subtracting the seasonal effect, then check the remainder.