Concept

# Autoregressive model

Summary
In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation which should not be confused with differential equation). Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable. Contrary to the moving-average (MA) model, the autoregressive model is not always stationary as it may contain a unit root. The notation indicates an autoregressive model of order p. The AR(p) model is defined as where are the parameters of the model, and is white noise. This can be equivalently written using the backshift operator B as so that, moving the summation term to the left side and using polynomial notation, we have An autoregressive model can thus be viewed as the output of an all-pole infinite impulse response filter whose input is white noise. Some parameter constraints are necessary for the model to remain weak-sense stationary. For example, processes in the AR(1) model with are not stationary. More generally, for an AR(p) model to be weak-sense stationary, the roots of the polynomial must lie outside the unit circle, i.e., each (complex) root must satisfy (see pages 89,92 ). In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model .
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (4)

Related people (4)
Related units

Related concepts

Related courses (20)
MATH-342: Time series
A first course in statistical time series analysis and applications.
FIN-403: Econometrics
The course covers basic econometric models and methods that are routinely applied to obtain inference results in economic and financial applications.
EE-512: Applied biomedical signal processing
The goal of this course is twofold: (1) to introduce physiological basis, signal acquisition solutions (sensors) and state-of-the-art signal processing techniques, and (2) to propose concrete examples
Related lectures (124)
Vector Autoregression: Modeling Vector-Valued Time Series
Explores Vector Autoregression for modeling vector-valued time series, covering stability, reverse characteristic polynomials, Yule-Walker equations, and autocorrelations.
Parametric Signal Models: Matlab Practice
Covers parametric signal models and practical Matlab applications for Markov chains and AutoRegressive processes.
Vector Autoregression
Explores Vector Autoregression for modeling vector-valued time series, covering stability, Yule-Walker equations, and spectral representation.
Related MOOCs