AR time series model

Autoregressive time series model. Uses past terms of the time series (with coefficients) to define present/ future behavior.

$$y_t=a+{\beta }_1{y}_{t-1}+\cdots +{\beta }_p{y}_{t-p}+{ϵ}_t$$

or can be written using the lag operators as

$$(1-{\beta }_{1}L-{\beta }_2{L}^2-\cdots -{\beta }_p{L}^p)y_t=c+{ϵ}_t$$

Is a stationary time series if the roots to the characteristic polynomial,

$$f(x)=1-{\beta }_{1}x-{\beta }_2{x}^2-\cdots -{\beta }_p{x}^p$$

are outside the unit circle ($||x||<1$).

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References

@alexander2007 - chapter 11 @pfaff2008 - chapter 1