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    Home >> Support >> Documentation >> NumXL >> Reference Manual >> ARMA Analysis >> ARIMA Analysis

    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:.

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

    Where:

    • x_t is the original non-stationary output at time t.
    • y_t is the observed differenced (stationary) output at time t.
    • d is the integration order of the time series.
    • 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.di) and follow a Gaussian distribution (i.e. \Phi(0,\sigma^2))

    notes

    1. The variance of the shocks is constant or time-invariant.
    2. Assuming y_t (i.e. (1-L)^d x_t ) is a stationary process with a long-run mean of \mu, then taking the expectation from both sides, we can express \phi_o as follows:
      \[ \phi_o = (1-\phi_1-\phi_2-\cdots -\phi_p)\mu \]
    3. Thus, the ARIMA(p,d,q) process can now be expressed as:
      \[ (1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) (y_t-\mu) = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t \]

      \[ z_t=y_t-\mu \]

      \[ (1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) z_t = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t \]
    4. In sum, z_t is the differenced signal after we subtract its long-run average.
    5. 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.
    6. Example: Consider the following ARIMA(12,2) process:

      \[ (1-\phi_1 L -\phi_{12} L^{12} ) (y_t-\mu) = (1+\theta_2 L^2 ) a_t \]

    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

    Related Links

    • Wikipedia - Autoregressive moving average model
    • ARIMA
    • ARIMA_CHECK
    • ARIMA_FIT
    • ARIMA_FORE
    • ARIMA_GOF
    • ARIMA_PARAM
    • ARIMA_SIM
    ‹ ARMA_FORECIupARIMA ›

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