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ARIMA Analysis
The ARIMAi model is an extension of the ARMAi 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:.
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Where:
-
is the original non-stationary output at time t.
-
is the observed differenced (stationary) output at time t.
-
is the integration order of the time series.
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.
)
notes
- The variance of the shocks is constant or time-invariant.
- Assuming
(i.e.
) is a stationary process with a long-run mean of
, then taking the expectation from both sides, we can express
as follows:
- Thus, the ARIMA(p,d,q) process can now be expressed as:
- In sum,
is the differenced signal after we subtract its long-run average.
- 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.
-
Example: Consider the following ARIMA(12,2) process:
Files Examples
References
- 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