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Econometrics and Financial Time Series / Univariate time series

AR, MA, and ARMA

A mean-reversion or short-rate forecast needs a dependence structure that remains useful across time. Stationarity provides that model premise. ARMA uses the series' history to construct a best linear forecast under a correctly specified stationary model. The forecast equals the conditional mean with additional assumptions, such as jointly Gaussian observations. This lesson explains model identification, root conditions, residual checks, and forecast mechanics. Unit roots and volatility clustering require separate checks.
Learning objectives. By the end of this note you should be able to:
  • distinguish strict from covariance stationarity and define white noise, ACF, and PACF;
  • write AR, MA, and ARMA in lag-operator form and state Wold's representation claim;
  • derive AR(1) stationarity and MA(1) invertibility and state the general root tests;
  • execute the Box–Jenkins loop with AIC/BIC and Ljung–Box;
  • compute AR(1) and MA(1) conditional forecasts and state briefly how forecast error variance grows with horizon.

In this lesson

  1. Stationarity, white noise, ACF, PACF
  2. AR, MA, ARMA, and Wold
  3. Stationarity and invertibility
  4. Box–Jenkins
  5. Forecasting mechanics (brief)

Before you start

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