ARIMA

ARIMA stands for AutoRegressive Integrated Moving Average.

The ARIMA Time Series Analysis uses lags and shifts in the historical data to uncover patterns (e.g. moving averages, seasonality) and predict the future. The ARIMA model was first developed in the late 60s but it was systemized by Box and Jenkins in 1976. ARIMA can be more complex to use than other statistical forecasting techniques, although when implemented properly ARIMA can be quite powerful and flexible.

ARIMA is a generalization of an autoregressive moving average (ARMA) model. These models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). They are applied in some cases where data show evidence of non-stationarity, where an initial differencing step (corresponding to the “integrated” part of the model) can be applied to remove the non-stationarity.

The model is generally referred to as an ARIMA(p,d,q) model where parameters p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively. ARIMA models form an important part of the Box-Jenkins approach to time-series modelling.

ARIMA is a method for determining two things:

  1. How much of the past should be used to predict the next observation (length of weights)
  2. The values of the weights.

For example y(t)= 1/3 * y(t-3) + 1/3 * y(t-2) + 1/3 * y(t-1) is an ARIMA model; another ARIMA MODEL is y(t)= 1/6 * y(t-3) + 4/6 * y(t-2) + 1/6 * y(t-1) Thus the correct ARIMA model requires identification of the right number of lags and the coefficients that should be used. ARIMA model identification uses autoregressions to identify the underling model. Care must be taken to robustly identify and estimate parameters as outliers (pulses, level shifts , local time trends ) can wreak havoc.

Source: Alan Pankratz – Forecasting with Univariate Box Jenkins Models : Concepts and Cases

Source: Jeffrey Wooldridge – Introductory Econometrics: A Modern Approach

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