ARIMA prototypes

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.