Autoregressive–moving-average model

Autoregressive–Moving-Average Model

The autoregressive–moving-average model (ARMA) is a statistical model used for analyzing and forecasting time series data. It is a combination of two models: the autoregressive (AR) model and the moving-average (MA) model. The ARMA model is widely used in the field of econometrics, finance, and various other disciplines that involve time series analysis.

Overview

The ARMA model is designed to capture the linear dependencies in a time series. It is defined by two parameters: \( p \) and \( q \), where \( p \) is the order of the autoregressive part and \( q \) is the order of the moving-average part.

Autoregressive (AR) Part

The autoregressive part of the model specifies that the output variable depends linearly on its own previous values. The AR part of an ARMA model of order \( p \) is given by:

\[ X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \varepsilon_t \]

where:

• \( X_t \) is the time series value at time \( t \).
• \( c \) is a constant.
• \( \phi_1, \phi_2, \ldots, \phi_p \) are the parameters of the model.
• \( \varepsilon_t \) is white noise.

Moving-Average (MA) Part

The moving-average part of the model specifies that the output variable depends linearly on the current and various past values of a stochastic (white noise) term. The MA part of an ARMA model of order \( q \) is given by:

\[ X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \cdots + \theta_q \varepsilon_{t-q} \]

where:

• \( \mu \) is the mean of the series.
• \( \theta_1, \theta_2, \ldots, \theta_q \) are the parameters of the model.

ARMA Model

Combining both parts, the ARMA model is expressed as:

\[ X_t = c + \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q} \]

Applications

ARMA models are used in various fields for:

• Forecasting: Predicting future values of a time series based on past observations.
• Signal Processing: Filtering and analyzing signals in engineering.
• Econometrics: Modeling economic and financial time series data.

Limitations

ARMA models assume that the time series is stationary, meaning its statistical properties do not change over time. If the series is non-stationary, it may need to be differenced or transformed before applying an ARMA model.

Also see

• Autoregressive model
• Moving-average model
• ARIMA model
• Time series analysis
• Stationary process