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.

ARIMA Model

The ARIMA (autoregressive integrated moving average) is a model describing time series
yt = 1,2…= {y1,y2,…}.


The full (p,d,q)(P,D,Q)s notation is given in the form of backshift polynomials. It encompasses set of simpler models:


The differencing model DIFF(d) of order d:

The seasonal differencing model DIFF(D)s of order D and seasonality lag s:

The autoregressive model AR(p) of order p:

The seasonal autoregressive model AR(P)s of order P and seasonality lag s:

The moving average model MA(q) of order q:

The seasonal moving average model MA(Q)s of order Q and seasonality lag s:

The εt is white noise (with properties: zero mean, constant variance, independence and normality):