ARIMA (Box-Jenkins Models): Autoregressive Integrated Moving Average

Time Series >

ARIMA modeling (sometimes called Box-Jenkins modeling), is an approach to modeling ARIMA processes—mathematical models used for forecasting. The approach uses previous time series data plus an error to forecast future values. More specifically, it combines a general autoregressive model AR(p) and general moving average model MA(q):

  • AR(p)— uses previous values of the dependent variable to make predictions.
  • MA(q)—uses the series mean and previous errors to make predictions.

The approach was first proposed by Box and Jenkins (1970), who detailed ARIMA’s estimation and prediction procedures (Hyndman et al., 2008).


Nonseasonal Autoregressive Integrated Moving Average models are classified by three factors:

  • p = number of autoregressive terms,
  • d = how many nonseasonal differences are needed to achieve stationarity,
  • q = number of lagged forecast errors in the prediction equation.

For example, an ARIMA(1,0,0) has 1 autoregressive term, no needed differences for stationarity and no lagged forecast errors. It is a special case of an ARIMA called a first-order autoregressive model.


The basic steps are (Hyndman, 2001):

  • Prepare your data by using transformations (e.g. square roots or logarithms) to stabilize the variance and differencing to remove remaining seasonality or other trends.
  • Identify any processes that appear to be a good fit for your data.
  • Find which model coefficients provide the best fit for your data. This step is computationally complex and usually performed by a computer. Akaike’s Information Criterion (AIC) is one option: if you compare two models, the one with the lower AIC is usually the “better” model.
  • Test the models’ assumptions to see how well the model holds up to closer scrutiny. If your chosen model is inadequate, repeat steps 2 and 3 to find a potentially better model.
  • Compute forecasts on your chosen model with computer software.


ARIMA models work on the assumption of stationarity (i.e. they must have a constant variance and mean). If your model is non-stationary, you’ll need to transform it before you can use ARIMA.


Box, G and Jenkins, G. (1970) Time series analysis: Forecasting and control, San
Francisco: Holden-Day.
Hyndman, R. (2001). Box-Jenkins modelling. Retrieved February 25, 2018 from:
Hyndman, R. et al. (2008). Forecasting with Exponential Smoothing: The State Space Approach. Springer Science & Business Media.
Nao, R. Introduction to ARIMA models. Retrieved February 27, 2018 from:

Comments? Need to post a correction? Please Contact Us.