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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 a 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.

Although Large Language Models are called autoregressive, they are not a classical autoregressive model in this sense because they are not linear.

Definition

The notation ${\displaystyle AR(p)}$ indicates an autoregressive model of order p. The AR(p) model is defined as

${\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}$

where ${\displaystyle \varphi _{1},\ldots ,\varphi _{p}}$ are the parameters of the model, and ${\displaystyle \varepsilon _{t}}$ is white noise.^[1][2] This can be equivalently written using the backshift operator B as

${\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}B^{i}X_{t}+\varepsilon _{t}}$

so that, moving the summation term to the left side and using polynomial notation, we have

${\displaystyle \phi X_{t}=\varepsilon _{t}}$

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 ${\displaystyle |\varphi _{1}|\geq 1}$ are not stationary. More generally, for an AR(p) model to be weak-sense stationary, the roots of the polynomial ${\displaystyle \Phi (z):=\textstyle 1-\sum _{i=1}^{p}\varphi _{i}z^{i}}$ must lie outside the unit circle, i.e., each (complex) root ${\displaystyle z_{i}}$ must satisfy ${\displaystyle |z_{i}|>1}$ (see pages 89,92 ^[3]).

Intertemporal effect of shocks

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 ${\displaystyle X_{t}=\varphi _{1}X_{t-1}+\varepsilon _{t}}$. A non-zero value for ${\displaystyle \varepsilon _{t}}$ at say time t=1 affects ${\displaystyle X_{1}}$ by the amount ${\displaystyle \varepsilon _{1}}$. Then by the AR equation for ${\displaystyle X_{2}}$ in terms of ${\displaystyle X_{1}}$, this affects ${\displaystyle X_{2}}$ by the amount ${\displaystyle \varphi _{1}\varepsilon _{1}}$. Then by the AR equation for ${\displaystyle X_{3}}$ in terms of ${\displaystyle X_{2}}$, this affects ${\displaystyle X_{3}}$ by the amount ${\displaystyle \varphi _{1}^{2}\varepsilon _{1}}$. Continuing this process shows that the effect of ${\displaystyle \varepsilon _{1}}$ never ends, although if the process is stationary then the effect diminishes toward zero in the limit.

Because each shock affects X values infinitely far into the future from when they occur, any given value X_t is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression

${\displaystyle \phi (B)X_{t}=\varepsilon _{t}\,}$

(where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as

${\displaystyle X_{t}={\frac {1}{\phi (B)}}\varepsilon _{t}\,.}$

When the polynomial division on the right side is carried out, the polynomial in the backshift operator applied to ${\displaystyle \varepsilon _{t}}$ has an infinite order—that is, an infinite number of lagged values of ${\displaystyle \varepsilon _{t}}$ appear on the right side of the equation.

Zdroj:https://en.wikipedia.org?pojem=AR(1)
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