ARMA Analysis

By definition, auto-regressive moving average (ARMA) is a stationary stochastic process made up of sums of auto-regressive Excel and moving average components.

Alternatively, in a simple formulation for an ARMA(p,q):

\[ x_t -\phi_o - \phi_1 x_{t-1}-\phi_2 x_{t-2}-\cdots -\phi_p x_{t-p}=a_t + \theta_1 a_{t-1} + \theta_2 a_{t-2} + \cdots + \theta_q a_{t-q}\]


  • \(x_t\) is the observed output at time t.
  • \(a_t\) 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. \(\Phi(0,\sigma^2)\)

Using back-shift notations (i.e. L), we can express the ARMA process as follows:

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

Assuming \(y_t\) is stationary with a long-run mean of \(\mu\), then taking the expectation from both sides, we can express \(\phi_o\) as follows:

\[ \phi_o = (1-\phi_1 -\phi_2 - \cdots - \phi_p)\mu \]

Thus, the ARMA(p,q) process can now be expressed as

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

In sum, \(z_t\) is the original signal after we subtract its long-run average.

  1. The variance of the shocks is constant or time-invariant.
  2. The order of an AR component process is solely determined by the order of the last lagged auto-regressive variable with a non-zero coefficient (i.e. \(w_{t-p}\).
  3. The order of an MA component process is solely determined by the order of the last moving average variable with a non-zero coefficient (i.e. \(a_{t-q}\).
  4. In principle, you can have fewer parameters than the orders of the model.
  5. Example: Consider the following ARMA(12,2) process: \[(1-\phi_1 L -\phi_{12} L^{12} )(y_t - \mu) = (1+\theta L^2)a_t\]
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