The ARIMA model is an extension of the ARMA model that applies to non-stationary time series (the kind of time series with one or more integrated unit-roots).By definition, the auto-regressive integrated moving average (ARIMA) process is an ARMA process for the differenced time series:.

\[ (1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p)(1-L)^d x_t - \phi_o= (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t\] \[ y_t = (1-L)^d x_t \]


  • \(x_t\) is the original non-stationary output at time t.
  • \(y_t\) is the observed differenced (stationary) output at time t.
  • \(d\) is the integration order of the time series.
  • \(a_t\) is the innovation, shock or error term at time t.
  • \(p\) is the order of the last lagged variables.
  • \(q\) is the order of the last lagged innovation or shock.
  • \(\{a_t\}\) time series observations are independent and identically distributed (i.e. i.i.d) and follow a Gaussian distribution (i.e. \(\Phi(0,\sigma^2)\)
  1. The variance of the shocks is constant or time-invariant.
  2. Assuming \(y_t\) (i.e. \((1-L)^d x_t\) ) is a stationary process with a long-run mean of \mu, then taking the expectation from both sides, we can express \(\phi_o\) as follows: \[\phi_o = (1-\phi_1-\phi_2-\cdots -\phi_p)\mu \]
  3. Thus, the ARIMA(p,d,q) process can now be expressed as: \[ (1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) (y_t-\mu) = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t \] \[z_t=y_t-\mu\] \[(1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) z_t = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t\]
  4. In sum, \(z_t\) is the differenced signal after we subtract its long-run average.
  5. The order of an ARIMA process is solely determined by the order of the last lagged variable with a non-zero coefficient. In principle, you can have fewer number of parameters than the order of the model.
  6. Example: Consider the following ARIMA(12,2) process:

    \[ (1-\phi_1 L -\phi_{12} L^{12} ) (y_t-\mu) = (1+\theta_2 L^2 ) a_t\]
Header SFSDK.H
* Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
* Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
* D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
* Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848