Applications of Time Series Models: AR, MA, ARMA and ARIMA

What this chapter covers

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

1. 1Time Series Components and StationarityEvery later model rests on the idea of stationarity, so you need it and the autocovariance and ACF definitions first.
2. 2AR, MA and ARMA ModelsThese are the basic building blocks, and you need their stationarity and invertibility conditions before you can difference or fit anything.
3. 3ARIMA Models and DifferencingARIMA extends ARMA to non-stationary series by differencing, so you can only learn it once ARMA is solid.
4. 4Model Fitting, Diagnostics and Box-Jenkins MethodOnce you know the model types, you learn how to choose between them using the ACF, PACF, estimation and residual checks.
5. 5Forecasting with Time Series ModelsForecasting uses a fitted model, so it comes after you can write down and estimate one.
6. 6Actuarial Applications and Multivariate ModelsThis ties the tools to real actuarial problems and extends them to several series, which is easier once the single-series case is clear.

How to prepare Applications of time series models

1. Learn the definitions of stationarity, autocovariance and the ACF, and write them out from memory until they are automatic.
2. For AR(p), MA(q) and ARMA models, practise checking stationarity and invertibility using the characteristic polynomial and its roots.
3. Practise deriving the ACF of AR(1), MA(1) and MA(2) by hand, and learn the shape each model gives for the ACF and PACF.
4. Work through the Box-Jenkins steps on a set of sample correlograms, naming the model you would try and the reason.
5. Do forecasting questions step by step: write the model, substitute known values, replace future errors with zero, and state the forecast variance where asked.
6. Repeat the fitting, diagnostics and forecasting in R so you can read the output, then finish with full past-paper questions under time.

Common mistakes in Applications of time series models

• Checking stationarity using the coefficients instead of the roots.

  Fix: Write the characteristic polynomial and check that all its roots lie outside the unit circle. For AR(1) this reduces to |α| < 1.

• Mixing up the ACF and PACF patterns for AR and MA models.

  Fix: Remember that the order of an AR model shows in the PACF cut-off, and the order of an MA model shows in the ACF cut-off.

• Forgetting the invertibility condition for MA and ARMA models.

  Fix: State both conditions separately: stationarity for the AR part and invertibility for the MA part.

• Using the wrong number of differences or differencing a stationary series.

  Fix: Difference only when the series or its ACF suggests non-stationarity, and stop once the series looks stationary. Over-differencing adds unneeded structure.

• Keeping the future error terms in a forecast.

  Fix: Replace future errors by zero and use actual or forecast values for the series. Past errors are known and stay.

• Reporting R output without interpreting it.

  Fix: After each output, write one sentence on what it shows, such as whether residuals are white noise or which model fits better.

Last-day revision: Applications of time series models

• ✓A weakly stationary series has constant mean, constant variance and autocovariance that depends only on the lag.
• ✓ACF at lag k is γk ÷ γ0, and ρ0 = 1.
• ✓AR(p) is stationary when all roots of its characteristic polynomial lie outside the unit circle.
• ✓MA(q) is always stationary, and it is invertible when the roots of its MA polynomial lie outside the unit circle.
• ✓For AR(p), the PACF cuts off after lag p and the ACF decays.
• ✓For MA(q), the ACF cuts off after lag q and the PACF decays.
• ✓ARIMA(p,d,q) means the series differenced d times is ARMA(p,q).
• ✓Box-Jenkins steps: identify, estimate, check the residuals, then forecast.
• ✓Good residuals look like white noise, with no significant autocorrelation.
• ✓In forecasts, future error terms are replaced by their expected value of zero.
• ✓Forecast variance grows as you forecast further ahead.
• ✓Differencing removes a trend, and seasonal differencing removes a seasonal pattern.

Applications of time series models practice questions

• Y_t follows an ARIMA(0,1,1) model: Y_t = Y_{t-1} + e_t + 0.4 e_{t-1}, with e_t white noise of variance 5. What is the variance of the differ…
• A stationary AR(1) process X_t = 0.5 X_{t-1} + e_t has e_t white noise with variance 9. What is the variance of X_t?
• Consider the AR(2) process X_t = 1.2 X_{t-1} - 0.2 X_{t-2} + e_t. Which statement is correct?
• An actuary fits an MA(1) model X_t = e_t + theta e_{t-1}. For the model to be invertible, which condition on theta is required?
• For a stationary AR(1) process X_t = 0.8 X_{t-1} + e_t with white-noise variance 3.6, what is the variance of X_t?
• Two series of annual rupee equity returns and bond returns for an Indian pension fund are each non-stationary, integrated of order 1. A line…
• The sample partial autocorrelation function of a stationary series of 400 observations shows significant spikes at lags 1 and 2 and is negli…
• An MA(1) model is X_t = mu + e_t + 0.5 e_{t-1}, with mu = 20. At time n the last observation is X_n = 23 and the estimated residual is e_n =…

Applications 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.

• CMA IntermediateData Analysis and Modelling
• CMA FinalBusiness Forecasting Models - Time Series and Regression Analysis
• CMA FoundationIndex Numbers and Time Series
• ACCA Applied KnowledgeSummarising and analysing data