Auto-Regressive Moving Average (ARMA) Model

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):

$$x_t -\phi_o - \phi_1 x_{t-1}-\phi_2 x_{t-2}-\cdots -\phi_p x_{t-p}=a_t + \theta_1 a_{t-1} + \theta_2 a_{t-2} + \cdots + \theta_q a_{t-q}$$

where:

  • $x_t$ is the observed output at time t.
  • $a_t$ 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.d) and follow a Gaussian distribution (i.e. $\Phi(0,\sigma^2)$)

Using back-shift notations (i.e. L), we can express the ARMA process as follows:

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

Assuming $y_t$ is stationary with a long-run mean of \mu, then taking the expectation from both sides, we can express $\phi_o$ as follows:

$$\phi_o = (1-\phi_1 -\phi_2 - \cdots - \phi_p)\mu$$ 

Thus, the ARMA(p,q) process can now be expressed as

$$(1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p) (x_t - \mu)= (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t$$

$$z_t = x_t - \mu$$

$$(1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p) z_t = (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t$$

In sum, $z_t$ is the original signal after we subtract its long-run average.

notes

  1. The variance of the shocks is constant or time-invariant.
  2. 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. $w_{t-p}$).
  3. 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. $a_{t-q}$).
  4. In principle, you can have fewer parameters than the orders of the model.

    Example: Consider the following ARMA(12,2) process:

    $$(1-\phi_1 L -\phi_{12} L^{12} )(y_t - \mu) = (1+\theta L^2)a_t$$

 

References

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