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ARMAX Analysis
In principle, an ARMAX model is a linear regression model that uses an ARMA^{i}type process (i.e. )to model residuals:.
Where:
 is the lag^{i} (aka backshift) operator.
 is the observed output at time t.
 is the kth exogenous input variable at time t.
 is the coefficient value for the kth exogenous (explanatory) input variable.
 is the number of exogenous input variables.
 is the autocorrelated regression residuals.
 is the order of the last lagged variables.
 is the order of the last lagged innovation or shock.
 is the innovation, shock or error term at time t.
 time series observations are independent and identically distributed (i.e. i.i.d) and follow a Gaussian distribution (i.e. )
Assuming and all exogenous input variables are stationary, then taking the expectation from both sides, we can express as follows:
is the longrun average of the ith exogenous input variable.
In the event that is not stationary, then one must verify that: (a) ane or more variables in is not stationary and (b) the time series variables in are cointegrated, so there is at least one linear combination of those variables that yields a stationary process (i.e. ARMA).
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