Core Concepts of Time Series Models for IAI Actuarial CS2

What this chapter covers

Core concepts of time series models: topics in the order to study them

1. 1Stationarity and Weak Stationarity of Time SeriesEvery later model is judged by whether it is stationary, so you need the definitions, the autocovariance function and the ACF first.
2. 2White Noise and Random Walk ProcessesThese are the simplest processes and the building blocks of AR, MA and ARIMA models; the random walk is also your first non-stationary example.
3. 3Autoregressive (AR) ModelsAR models introduce the backshift operator, the stationarity condition on the characteristic roots and the Yule-Walker equations.
4. 4Moving Average (MA) Models and InvertibilityMA models mirror AR models, with a finite ACF and the invertibility condition, so they are easier once AR is clear.
5. 5ARMA and ARIMA ModelsThese combine AR and MA parts, and ARIMA adds differencing to handle non-stationary data.
6. 6Model Identification, Fitting and ForecastingThis topic uses everything before it: you pick a model from the ACF and PACF, estimate it, check residuals and forecast.

How to prepare Core concepts of time series models

1. Write out the definitions of weak stationarity: constant mean, variance finite and constant, and autocovariance depending only on the lag. Practise testing simple processes against them.
2. Learn the backshift operator and rewrite each model as a polynomial in B. Practise finding roots and checking whether they lie outside the unit circle.
3. Derive the ACF of AR(1), MA(1) and, where possible, AR(2) and MA(2) by hand until the steps feel routine. Use Yule-Walker equations for AR models.
4. Build a one-page table of the ACF and PACF patterns: which one cuts off, which one tails off, for AR, MA and ARMA.
5. Practise forecasting: compute one-step and multi-step forecasts, and see how forecasts of a stationary model move towards the mean as the horizon grows.
6. Repeat the fitting cycle in R on a sample data set: plot, difference if needed, inspect ACF and PACF, fit with arima, check residuals and forecast.
7. Finish with past-paper style questions, writing out your working in full, with assumptions stated.

Common mistakes in Core concepts of time series models

• Checking stationarity using the roots of the wrong polynomial or the wrong region.

  Fix: Pick one convention and stick to it. In the B form, roots must lie outside the unit circle, which means |B| > 1.

• Confusing the stationarity condition for AR with the invertibility condition for MA.

  Fix: Remember that AR models need a stationarity check and MA models need an invertibility check. MA models are always stationary.

• Reading the ACF and PACF the wrong way round when identifying a model.

  Fix: Use your table: PACF cuts off for AR, ACF cuts off for MA, and both tail off for ARMA. Say it aloud before every identification.

• Forgetting to difference before fitting a series with a clear trend, or over-differencing.

  Fix: Difference only until the series looks stationary, usually once or twice. Over-differencing adds unwanted MA structure.

• Dropping the mean when writing or forecasting a model.

  Fix: Work with the deviation Xt − μ, forecast it, and add μ back at the end.

• Giving a bare answer without stating assumptions or showing steps.

  Fix: State the model, the assumptions on the noise term, each step of the working and the final result with its interpretation.

Last-day revision: Core concepts of time series models

• ✓Weak stationarity: constant mean, constant finite variance, and covariance that depends only on the lag.
• ✓Autocorrelation at lag k: ρk = γk ÷ γ0, and ρ0 = 1.
• ✓White noise: uncorrelated terms with zero mean and constant variance.
• ✓Random walk: Xt = Xt-1 + et is non-stationary, but its first difference is white noise.
• ✓AR(p) is stationary when all roots of its characteristic polynomial lie outside the unit circle.
• ✓AR(1): Xt = μ + α(Xt-1 − μ) + et is stationary if |α| < 1, with ρk = α^k.
• ✓MA(q) is always stationary and its ACF is zero beyond lag q.
• ✓MA(q) is invertible when all roots of its MA polynomial lie outside the unit circle.
• ✓AR has a PACF that cuts off after lag p; MA has an ACF that cuts off after lag q.
• ✓ARIMA(p,d,q) means the series differenced d times follows an ARMA(p,q) model.
• ✓Stationary forecasts tend to the mean as the horizon grows; random-walk forecasts do not.
• ✓Check residuals: they should look like white noise, with no remaining ACF pattern.

Core concepts of time series models practice questions

• A time series X_t satisfies X_t = 0.6 X_{t-1} + e_t, where e_t is white noise. Which statement about this process is correct?
• A stationary AR(1) process has autocorrelation at lag 1 equal to 0.5. What is its autocorrelation at lag 3?
• For a stationary MA(1) process X_t = e_t + 0.5 e_{t-1}, with e_t white noise of variance sigma^2, what is the autocorrelation at lag 1 and a…
• Which of the following processes, with e_t independent white noise of constant variance, is NOT weakly stationary?
• Consider the process X_t = 1.5 X_{t-1} - 0.5 X_{t-2} + e_t, where e_t is white noise. Which description is correct?
• Consider the process X_t = 0.5 X_{t-1} + e_t, where e_t are independent N(0, 4) white noise terms, and the process has been running for a lo…
• If X_t follows a random walk X_t = X_{t-1} + e_t with e_t white noise, which transformation produces a stationary series?
• A series shows a clear upward trend and a sample ACF that decays very slowly from values near 1. The first differences of the series have an…

Core concepts of time series models in other exams

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

• FRM Part IStationary Time Series
• CFA Level IITime-Series Analysis