Autoregressive Integrated Moving Average - Definition

Definition

Given a time series of data where is an integer index and the are real numbers, then an ARMA(p,q) model is given by:

\left( 1 - \sum_{i=1}^{p} \alpha_i L^i \right) X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,

where is the lag operator, the are the parameters of the autoregressive part of the model, the are the parameters of the moving average part and the are error terms. The error terms are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.

Assume now that the polynomial has a unitary root of multiplicity d. Then it can be rewritten as:

\left( 1 - \sum_{i=1}^p \alpha_i L^i \right) = \left( 1 - \sum_{i=1}^{p-d} \phi_i L^i \right) \left( 1 - L \right)^{d} .


An ARIMA(p,d,q) process expresses this polynomial factorisation property, and is given by:

\left( 1 - \sum_{i=1}^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,

and thus can be thought as a particular case of an ARMA(p+d,q) process having the auto-regressive polynomial with some roots in the unity. For this reason every ARIMA model with d>0 is not wide sense stationary.

Read more about this topic:  Autoregressive Integrated Moving Average

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