There are some interesting prototypic
ARIMA models. In fact the ARIMA framework is that much general that it covers whole list of time series models:
ARIMA(0,0,0) is white noise (with optional level based on
c):
ARIMA(0,1,0) is random walk (with opt. drift based on
c):
ARIMA(1,1,0) is differenced first-order autoregressive model (with opt. drift based on
c):
ARIMA(0,1,1) is exponential smoothing (with opt. drift based on
c):
ARIMA(0,2,2) is linear exponential smoothing (with opt. curvature based on
c):
ARIMA(1,1,2) is damped-trend linear exponential smoothing (with opt. drift based on
c):
ARIMA(0,0,0)(1,1,0)12 is annually differenced first-order autoregressive model (with opt. drift based on
c):
ARIMA(0,1,0)(0,1,0)12 is annual random trend model (with opt. curvature based on
c):
It is also interesting to show that
AR and MA terms can mimic differencing (more on this in
Robert Nau: Statistical forecasting: notes on regression and time series analysis). It's easiest to explain on
ARIMA(1,1,1) model:
It is evident, that if
ϕ→1 it mimics the differencing
(1-B) and, on the contrary, if
θ→1 it mimics integrating, as the
(1-B) on both sides of equation “cancel each” other. In a similar way if
ϕ≈θ the
(1- ϕB) “cancels”
(1- θB) out
.
It is also shown that a
MA model can be expressed as infinite AR model (and vice versa). Consider simple
MA(1):
Then with the “hack” of infinite geometric series expression:
Possible lesson from the “mimicking” topic would be: do not “overfit” by trying model where simultaneously both (
AR and
MA) term orders are 2 or more (e.g.
ARIMA(2,1,2)) because the terns might cancel each other out.
