ARMAX (Pro.)

Returns an array of cells for the packed form of a given ARMAX model.

Syntax

ARMAX(mean, sigma, phi, theta, Beta)

mean
is the ARMA mean (i.e. mu).

sigma
is the standard deviation of the model's residuals.

phi
are the parameters of the AR(p) component model (starting with the lowest lag).

theta
are the parameters of the MA(q) component model (starting with the lowest lag).

Beta
are the coefficients array of the exogenous factors.

The time series is homogeneous or equally spaced.
The ARMAX model is written as:

$X_t = \mu + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{j=1}^q \theta_j a_{t-j} +\sum_{k=1}^N{\beta_k X_t^{(k)}} + a_t$

Where:
- $X_{t}$ is the value of the time series at time t.
- $\mu$ is the model mean.
- $p$ is the order of the autoregressive (AR) component model.
- $\phi_1,\phi_2,...,\phi_p$ are the AR(p) model parameters.
- $q$ is the order of the moving-average (MA) component model.
- $\theta_1,\theta_2,...,\theta_q$ are the MA(q) model parameters.
- $\beta_1,\beta_2,...,\beta_N$ are the coefficients for the exogenous factors.
- $X_t^{(k)}$ is the value of the k-th exogenous factor at time t.
- $a_t$ is the model's residual/innovation at time t.
  
  $\left[a_t\right] \sim i.i.d$
  
  $a_t \sim N(0,\sigma^2)$
The number of parameters in the input argument - phi - determines the order of the AR component.
The number of parameters in the input argument - theta - determines the order of the MA component.
The residuals/innovations standard deviation (i.e. sigma) should be greater than zero for stable ARMA model .
The ARMAX function returns an array of size :
1. $6$
2. $p$
3. $q$
4. $\mu$
5. $\sigma$
6. $\phi_1,\phi_2,...,\phi_p$
7. $\theta_1,\theta_2,...,\theta_q$
8. $\beta_1,\beta_2,...,\beta_N$

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

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