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autoregressive model


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A model for a time series having no trend (the constant mean is taken as 0). Let X1, X2, …, be successive instances of the random variable X, measured at regular intervals of time. Let εj be the random variable denoting the random error at time j. A pth-order autoregressive model (or autoregressive process) relates the value at time j to the preceding p values by

Xj=α1Xj−1+α2Xj−2+…+αpXj−p+εj,

where α1, α2,…, αp are constants. Such a model is written in brief as AR(p). The AR(1) process is a Markov chain. Autoregressive models can also be expressed as moving average models. Models combining both type of process include ARMA models and ARIMA models.

The Yule–Walker equations, introduced by Yule in 1927 and Walker in 1931, relate α1, α2,…, αp to the population autocorrelation values ρ1, ρ2,…, ρp:

ρ1=α1+α2ρ1+…+αpρp−1,

ρ2=α1ρ1+α2+…+αpρp−2,

ρp=α1ρp−1+α2ρp−2+…+αp.

Subjects: Probability and Statistics.


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