READING 27: MODELING CYCLES: MA, AR, AND ARMA MODELS

First-order Moving Average (MA(1)) Process

        The first-order moving average process enables forecasters to consider the likely current effect on a dependent variable of current and lagged white noise error terms. While this is a useful process, it is most useful when inverted as an autoregressive representation so that current observables can be explained in terms of past observables.

General Finite-order Process(MA(q))

        While the first-order moving average process does provide useful information for forecasting, the q^th-order moving average process allows for a richer analysis because it incorporates significantly more lagged error terms all the way out to the order of q.

First-order Autoregressive (AR(1)) Process

        The first-order autoregressive process incorporates the benefits of an inverted MA(1) process. Specifically, the AR(1) process seeks to explain the dependent variable in terms of a lagged observation of itself and an error term. This is a better forecasting tool if the autocorrelations decay gradually rather than cut off immediately after the first observation with a first-order process.

General p^th order Autoregressive (AR(p)) Process

        The pth-order autoregressive process adds additional lagged observations of the dependent variable and enhances the informational value relative to an AR(1) process in much the same way that an MA(q) process adds a richer explanation to the MA(1) process.

Autoregressive Moving Average (ARMA) Process

        The autoregressive moving average (ARMA) process has the potential to capture more robust relationships. The ARMA process incorporates the lagged error elements of the moving average process and the lagged observations of the dependent variable from the autoregressive process.

Application of AR and ARMA processes

        Both autoregressive (AR) and autoregressive moving average (ARMA) processes can be applied to time series data that show signs of seasonality. Seasonality is most apparent when the autocorrelations for a data series do not abruptly cut off, but rather decay gradually with periodic spikes.