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):
- 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.di) and follow a Gaussian distribution (i.e. )
Using back-shift notations (i.e. ), we can express the ARMA process as follows:
Assuming is stationary with a long-run 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 long-run average.
- The variance of the shocks is constant or time-invariant.
- 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. ).
- 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. ).
- In principle, you can have fewer parameters than the orders of the model.
Example: Consider the following ARMA(12,2) process: