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FRM Part I · FRM Exam Part I

Stationary Time Series for FRM Part I: Chapter Guide

A stationary time series has a stable mean, a stable variance and autocovariances that depend only on the lag, not on time. To solve questions, identify the process (white noise, AR, MA or ARMA), apply its mean, variance and autocorrelation formulas, then check stationarity conditions such as |φ| < 1 for an AR(1).

What this chapter covers

This chapter in Quantitative Analysis teaches you how to model data that moves through time, such as returns, spreads and volatility. The core idea is covariance stationarity. If a series has a constant mean, constant variance and autocovariances that depend only on the lag, you can model and forecast it with a small set of building blocks.

Those blocks are white noise, autoregressive (AR) models, moving average (MA) models and their combination, ARMA. You learn what each one implies for the autocorrelation function (ACF) and partial autocorrelation function (PACF). You then learn how to pick a model, estimate it and produce forecasts. The chapter ends with seasonality, which you handle with seasonal terms or dummy variables.

The chapter links to the rest of the paper in several ways. It builds on probability, regression and hypothesis testing earlier in Quantitative Analysis. It supports volatility and correlation modelling, such as EWMA and GARCH, and it feeds into market risk and VaR work in Valuation and Risk Models. Learn it well and those later topics become easier.

The exam has 100 equally weighted multiple-choice questions in 4 hours, and this chapter gives you questions that are short, formula-driven and quick to score once you know the patterns. Typical tasks are computing the mean or variance of an AR(1), reading ACF and PACF shapes, checking stationarity, and producing a one- or two-step forecast. Candidates who skip the chapter lose easy marks. Candidates who learn the few formulas well gain them reliably. The ideas also support later topics, so the time you spend here pays off more than once.

Stationary Time Series: topics in the order to study them

  1. 1Covariance Stationary Time SeriesStart here because every later model is built on the stationarity conditions and the idea of autocovariance and autocorrelation.
  2. 2White Noise ProcessesWhite noise is the simplest stationary process and the error term in every AR and MA model, so you need it before the models.
  3. 3Autoregressive (AR) ModelsAR(1) is the most tested model; it introduces the stationarity condition, the long-run mean and geometric decay of the ACF.
  4. 4Moving Average (MA) ModelsMA models contrast with AR models: the ACF cuts off after lag q, and learning both side by side sharpens pattern recognition.
  5. 5ARMA ModelsARMA combines AR and MA terms, so you need both pieces first; its ACF and PACF both decay gradually.
  6. 6Model Selection, Estimation and ForecastingThis topic uses all the models above: you choose between them with ACF and PACF plots and information criteria, then forecast.
  7. 7Seasonality in Time SeriesSeasonality is a short add-on to the framework; it is easiest once you know how autocorrelation shows up at seasonal lags.

How to prepare Stationary Time Series

Plan for a few focused sessions. Work with formulas and small numerical examples rather than reading alone, because the exam rewards fast and accurate calculation.

  1. Write down the three conditions of covariance stationarity in your own words and test them on simple examples, such as a trending series and a constant-mean series.
  2. Memorise the white noise properties: mean zero, constant variance, no autocorrelation at any lag. Know that independent white noise is a stricter case than plain white noise.
  3. Work through AR(1) by hand: the stationarity condition |φ| < 1, the mean μ = c ÷ (1 − φ), the variance σ² ÷ (1 − φ²) and the autocorrelation at lag k, which equals φ^k.
  4. Draw the typical ACF and PACF shapes for AR(p), MA(q) and ARMA on one page. Practise matching a description or plot to a model until it takes seconds.
  5. Practise one-step and multi-step forecasts, and note how AR forecasts revert toward the mean while MA forecasts become the mean after q steps.
  6. Finish with mixed practice questions on model selection and seasonality, then redo the ones you missed after a day. Use a calculator for powers and square roots, and keep a formula sheet to rehearse.

Common mistakes in Stationary Time Series

  • Treating |φ| = 1 or larger as stationary for an AR(1).

    Fix: Check |φ| < 1 first. If it fails, the process is not covariance stationary and the mean and variance formulas do not apply.

  • Swapping the ACF and PACF patterns for AR and MA models.

    Fix: Remember that AR cuts off in the PACF and MA cuts off in the ACF. Rehearse the one-page sketch until it is automatic.

  • Using σ² instead of σ²(1 + θ²) for the variance of an MA(1).

    Fix: Add the squared coefficients of all shocks in the process, including the coefficient of 1 on the current shock.

  • Confusing white noise with independent white noise.

    Fix: White noise is uncorrelated. Independent white noise is independent over time, which is stronger, and normal white noise adds normality.

  • Forecasting an AR(1) multi-step ahead without the mean-reversion step.

    Fix: Substitute the previous forecast at each step and set future shocks to zero. The forecast moves toward the long-run mean.

  • Choosing the model with the lowest in-sample error and ignoring parameter penalties.

    Fix: Compare AIC or BIC, which penalise added parameters. Prefer the parsimonious model unless the extra terms clearly help.

Last-day revision: Stationary Time Series

  • Covariance stationary: constant mean, constant variance, autocovariance depends only on the lag.
  • White noise: mean 0, constant variance, zero autocorrelation at all lags.
  • AR(1): yt = c + φyt−1 + εt; stationary only if |φ| < 1.
  • AR(1) mean is c ÷ (1 − φ); variance is σ² ÷ (1 − φ²).
  • AR(1) autocorrelation at lag k is φ^k, so it decays geometrically.
  • MA(1): yt = μ + εt + θεt−1; always stationary; ACF is zero beyond lag 1.
  • MA(1) variance is σ²(1 + θ²).
  • AR(p): ACF decays gradually, PACF cuts off after lag p.
  • MA(q): ACF cuts off after lag q, PACF decays gradually.
  • ARMA: both ACF and PACF decay gradually.
  • Information criteria such as AIC and BIC penalise extra parameters; BIC penalises more heavily.
  • Seasonality shows as significant autocorrelation at seasonal lags and can be handled with seasonal dummies or seasonal terms.

Stationary Time Series practice questions

Stationary Time Series in other exams

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

Stationary Time Series: frequently asked questions

Do I need to derive the AR and MA formulas for the FRM exam?

No. Questions are multiple-choice, so you mainly need to apply the formulas correctly. Knowing where they come from helps you remember them, but you will not be asked for full derivations.

How do I tell an AR model from an MA model using plots?

Look at where the pattern stops. For an AR(p) process, the PACF cuts off after lag p while the ACF decays gradually. For an MA(q) process, the ACF cuts off after lag q while the PACF decays gradually.

Is an MA model always stationary?

A finite-order MA model is always covariance stationary, because it is a finite combination of white noise terms. An AR model needs a condition on its coefficients, such as |φ| < 1 for AR(1).

Which information criterion should I prefer, AIC or BIC?

Both penalise extra parameters, and BIC penalises more heavily, so it tends to select smaller models. On the exam, know that distinction and that a lower value indicates a preferred model.