ARIMA

ARIMA is an acronym for autoregressive, integrated, moving average, denoting the three components of a general class of stochastic time-series models described by Box and Tiao (1965, 1975) and developed by Box and Jenkins (1976). Estimated univariate ARIMA models are often used successfully for forecasting and also form the basic foundation for multivariate analysis, such as INTERVENTION ANALYSIS assessment or TRANSFER FUNCTION models in the BOX-JENKINS MODELING approach to time series. The use of ARIMA models in conjunction with transfer function models is often contrasted with the use of time-series regression models, but ARIMA models are increasingly used in tandem with regression models as well.

The Pieces

ARIMA models have three components, with each one describing a key feature of a given time-series process. Briefly, the three components correspond to the effect of past values, called an autoregressive (AR) process; past shocks, called a moving average (MA) process; and the presence of a stochastic trend, called an integrated (I) process. Each of these is discussed below with reference to example time-series processes.

Autoregressive

Time-series processes are often influenced by their own past values. Consider the percentage of the public that says they approve of the way the president is doing his job. Public sentiment about the president's performance today depends in part on how they viewed his performance in the recent past. More generally, a time series is autoregressive if the current value of the process is a function of past values of the process. The effect of past values of the process is represented by the autoregressive portion of the ARIMA model, written generally as follows:

Each observation consists of a random shock and a linear combination of prior observations. The number of lagged values determines the order of autoregression and is given by p, denoted AR(p).

Moving Average

ARIMA models assume that time-series processes are driven by a series of random shocks. Random shocks may be loosely thought of as a collection of inputs occurring randomly over time. More technically, random shocks are normally and identically distributed random variables with mean zero and constant variance. As an example, consider that reporting requirements for violent crime may change. This change “shocks” statistics about violent crime so that today's reports are a function of this (and other) shock(s). We can write this as follows:

where q gives the number of past shocks, μ that affect the current value of the y¯ process, here violent crime statistics. The number of lagged values of μ determines the moving average order and is given by q, denoted MA(q).

Integrated

ARIMA models assume that the effects of past shocks and past values decay over time: Current values of presidential approval reflect recent shocks and recent values of approval more so than distant values. For example, mistakes a president makes early in his tenure are forgotten over time. This implies that the parameter values in the general AR and MA model components must be restricted to ensure that effects decay (i.e., that the time series is stationary). Loosely, a stationary time series is one with constant mean, variance, and covariances over time. A series that is not stationary is known as integrated. Some economic time series (e.g., economic growth) are thought to be integrated because they exhibit a historical tendency to grow.

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