ARIMA Modeling

If we combine differencing with autoregression and a moving average model, we obtain a non-seasonal ARIMA model. ARIMA is an acronym for AutoRegressive Integrated Moving Average model. The term "integration" in this context is the reverse of differencing.

ARIMA model is represented as ARIMA(p,d,q)


  • p = order of the autoregressive part
  • d = degree of first differencing involved
  • q = order of the moving average part

A series which is stationary after being differentiated once is said to be integrated of order 1 and is denoted by I(1). In general a series which is stationary after being differentiated d times is said to be integrated of order d, denoted I(d). A series which is stationary without differencing is said to be I(0).

While using ARIMA modeling for time series forecasting, selecting appropriate values for p, d and q can be difficult. We make use of ACF, PACF and other functions to help us determine these values.

Many of the models we previously discussed can be easily explained by the ARIMA model as shown below:

  • White noise: ARIMA(0,0,0)
  • Random walk: ARIMA(0,1,0) with no constant
  • Random walk with drift: ARIMA(0,1,0) with a constant
  • Autoregression: ARIMA(p,0,0)
  • Moving average: ARIMA(0,0,q)

ARIMA Modeling in R

In the next few lessons, we will learn about how we can simulate and estimate these models in R using the ARIMA class of functions.

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