Arma

TL;DR

• Models current time series values using both past observed values and averages of recent values.
• Specified by two orders (p, q) — e.g., ARMA(2,1) or ARMA(1,3) — indicating numbers of AR and MA terms.
• Commonly applied to forecasting problems in domains with historical sequential data.

Definition

ARMA, or Autoregressive Moving Average, is a type of statistical model used in time series analysis to capture the effects of past values on current values in the data. It is a combination of two models: autoregression (AR) and moving average (MA).

Explanation

Autoregression (AR) regresses the dependent variable on its own lagged values, explaining current values of the data based on past values of the same data. An AR(1) model uses one lagged value; an AR(2) model uses the values at times t-1 and t-2 to predict the value at time t.

Moving average (MA), as described here, explains the current value of the data based on the average of past values. An MA(3) model uses the average of the last three values.

ARMA combines both autoregressive and moving average terms in a single model. For example, an ARMA(2,1) model includes two autoregressive terms and one moving average term, while an ARMA(1,3) model includes one autoregressive term and three moving average terms. This combination allows the model to capture both individual past values and averages of past values when predicting the current value.

Examples

Stock market prediction

Using historical daily stock prices for the past year, an ARMA(2,1) model can be built to predict the stock price at time t based on the stock prices at times t-1 and t-2, as well as the average of the stock prices at times t-1, t-2, and t-3.

Forecasting economic indicators

Using monthly unemployment rates for the past 10 years, an ARMA(1,3) model can be used to predict the unemployment rate at time t based on the unemployment rate at time t-1 and the average of the unemployment rates at times t-1, t-2, and t-3.

Use cases

• Finance (e.g., stock price prediction)
• Economics (e.g., forecasting unemployment rates, GDP)
• Meteorology (forecasting weather-related time series)
• General forecasting of sequential data based on historical values

Related terms

• Autoregression (AR)
• Moving average (MA)
• Time series analysis
• Forecasting