Have a Question?
Phone: +1 (888) 4279486
+1 (312) 2573777
Contact Us
ARMA Analysis
By definition, autoregressive moving average (ARMA) is a stationary stochastic process made up of sums of autoregressive Excel and moving average components.
Alternatively, in a simple formulation for an ARMA(p,q):
where:
 is the observed output at time t.
 is the innovation, shock or error term at time t.
 is the order of the last lagged variables.
 is the order of the last lagged innovation or shock.
 time series observations are independent and identically distributed (i.e. i.i.d^{i}) and follow a Gaussian distribution (i.e. )
Using backshift notations (i.e. ), we can express the ARMA process as follows:
Assuming is stationary with a longrun mean of , then taking the expectation from both sides, we can express as follows:
Thus, the ARMA(p,q) process can now be expressed as
In sum, is the original signal after we subtract its longrun average.
notes
 The variance of the shocks is constant or timeinvariant.
 The order of an AR component process is solely determined by the order of the last lagged autoregressive variable with a nonzero coefficient (i.e. ).
 The order of an MA component process is solely determined by the order of the last moving average variable with a nonzero coefficient (i.e. ).
 In principle, you can have fewer parameters than the orders of the model.

Example: Consider the following ARMA(12,2) process:
Files Examples
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740